Sonidegib and Lenalidomide After Stem Cell Transplant in Treating Patients With Multiple Myeloma

Inclusion Criteria:

* Absolute neutrophil count \>= 1500/uL
* Platelet count \>= 80,000/uL
* Hemoglobin \>= 9.0 g/dL
* Serum total bilirubin =\< 1.5 x ULN (upper limit of normal)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN or =\< 5 x ULN if liver involvement
* Plasma creatine phosphokinase (CK) \< 1.5 x ULN
* Serum creatinine =\< 1.5 x ULN or 24-hour clearance \>= 50 ml/min
* Diagnosis of symptomatic multiple myeloma (MM)
* Patients should have received single autologous stem cell transplantation 60-120 days prior to enrollment to the trial
* Patients should have received the autologous SCT within 12 months of their diagnosis of myeloma to be eligible for the study
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
* Recovered from toxicity of previous chemotherapy (excludes grade 1 neurotoxicity and hematological toxicity)
* Voluntary written informed consent before performance of any study-related procedure not part of normal medical care, with the understanding that the subject may withdraw consent at any time without prejudice to future medical care
* Willingness to return to the Mayo Clinic enrolling institution for follow-up
* Measurable disease of multiple myeloma at the time of baseline values for disease assessment as defined by at least one of the following:

  * Serum monoclonal protein \>= 1.0 g/dL
  * \>= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis
  * Serum immunoglobulin free light chain (involved free light chain \[FLC\]) \>=10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio

    * NOTE: For patients with no relapse prior to transplant, measurable disease at the time of diagnosis
    * NOTE: For patients who have had a disease relapse prior to transplant, measurable disease at the time of the most recent relapse immediately prior to transplant; NOTE: if the patient had treatment for the relapsed disease prior to transplant, the patient must have measurable disease at the time of relapse prior to this therapy
* Patients may have a history of current or previous deep vein thrombosis or pulmonary embolism but must be willing to initiate prophylaxis with low molecular weight heparin
* Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 50 mIU/mL within 10-14 days prior to and again within 24 hours of starting lenalidomide and LDE225 and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking lenalidomide and LDE225; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a successful vasectomy; all patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure; a female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: has not undergone a hysterectomy or bilateral oophorectomy; or has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)
* Sexually active males must be willing to use a condom (even if they have undergone a prior vasectomy) while having intercourse, while taking lenalidomide and for 4 weeks after stopping treatment
* Patient enrolling to this study must agree to register to the mandatory REVLIMID Risk Evaluation and Mitigation Strategy (REMS \[TM\]) program, and be willing and able to comply with the requirements of (REVLIMID REMS \[TM\])
* Patients must be willing to provide biological samples as required by the study

Exclusion Criteria:

* Prior allogeneic bone marrow/peripheral blood stem cell transplant
* Patients with evidence of disease progression post SCT at the time of consideration for the study enrollment will not be included
* Impaired cardiac function or clinically significant heart disease, including any one of the following:

  * Angina pectoris within 3 months
  * Acute myocardial infarction within 3 months
  * Fridericia QT (QTcF) \> 450 msec for males and \> 470 msec for females on the screening electrocardiogram (ECG)
  * A past medical history of clinically significant ECG abnormalities or a family history of prolonged QT-interval syndrome
  * Other clinically significant heart disease (e.g. congestive heart failure, uncontrolled hypertension, history of labile hypertension, or history of poor compliance with an antihypertensive regimen)
* Seroreactivity for human immunodeficiency virus (HIV), human T-lymphotropic virus type I (HTLV I) or II, hepatitis B virus (HBV), hepatitis C virus (HCV)
* Other active malignancy requiring therapy; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix
* Any of the following:

  * Pregnant or nursing (lactating) women, where pregnancy is defined as the state of a female after conception and until the termination of gestation, confirmed by a positive human chorionic gonadotropin (hCG) laboratory test (\> 5 mIU/mL)
  * Patients who are not willing to apply highly effective contraception during the study and through the duration as defined below after the final dose of study treatment
  * Women of child-bearing potential, defined as all women physiologically capable of becoming pregnant, must use highly effective contraception during the study and through 6 months after the final dose of study treatment; highly effective contraception is defined as either:

    * Total abstinence: when this is in line with the preferred and usual lifestyle of the patient; periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception
    * Sterilization: patient has had surgical bilateral oophorectomy (with or without hysterectomy) or tubal ligation at least six weeks before taking study treatment; in case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment
    * Male partner sterilization (with the appropriate post-vasectomy documentation of the absence of sperm in the ejaculate); (for female study patients, the vasectomized male partner should be the sole partner for that patient)
    * Use a combination of the following (both a+b):

      * Placement of a non-hormonal intrauterine device (IUD) or non-hormonal intrauterine system (IUS)
      * Barrier method of contraception: condom or occlusive cap (diaphragm or cervical vault caps) with spermicidal foam/gel/film/cream/vaginal suppository
    * Note: hormonal contraception methods (e.g. oral, injected, implanted) are not allowed
    * Note: woman are considered post-menopausal and not child bearing potential if they have had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile (e.g. age appropriate, history of vasomotor symptoms) or six months of spontaneous amenorrhea with serum follicle stimulating hormone (FSH) levels \> 40 mIU/mL and estradiol \< 20 pg/mL or have had surgical bilateral oophorectomy (with or without hysterectomy) at least six weeks ago; in the case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment is she considered not of child bearing potential
  * Male patient must use highly effective (double barrier) methods of contraception (e.g., spermicidal gel plus condom) for the entire duration of the study, and continuing using contraception and refrain from fathering a child for 6 months following the study drug; a condom is required to be used also by vasectomized men as well as during intercourse with a male partner in order to prevent delivery of the study treatment via seminal fluid
  * Sexually active males who are unwilling to use a condom during intercourse while taking drug and for 6 months after stopping investigational medications and agree not to father a child in this period
* Other co-morbidity, which would interfere with patient's ability to participate in the trial, e.g. uncontrolled infection, uncompensated lung disease
* Concurrent chemotherapy, radiotherapy, or any ancillary therapy for treatment of multiple myeloma; NOTE: bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment
* Known allergies to any of the components of the investigational treatment regimen or required ancillary treatments
* Major surgery within 4 weeks prior to registration
* Patients with concurrent uncontrolled medical conditions that may interfere with their participation in the study or potentially affect the interpretation of the study data
* Patients unable to take oral drugs or with lack of physical integrity of the upper gastrointestinal tract or known malabsorption syndromes
* Patients who have previously been treated with systemic LDE225 or with other hedgehog pathway inhibitors
* Patients who have neuromuscular disorders (e.g. inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis and spinal muscular atrophy) or are on concomitant treatment with drugs that are recognized to cause rhabdomyolysis, such as 5-hydroxy-3-methylglutaryl-coenzyme A (HMG CoA) inhibitors (statins), clofibrate and gemfibrozil, and that cannot be discontinued at least 2 weeks prior to starting LDE225 treatment; NOTE: if it is essential that the patient stays on a statin to control hyperlipidemia, only pravastatin may be used with extra caution
* Patients who have taken part in an experimental drug study =\< 4 weeks prior to registration
* Patients who are receiving treatment with medications known to be moderate and strong inhibitors or inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4)/cytochrome P450, family 3, subfamily A, polypeptide 5 (5) or drugs metabolized by cytochrome P450, family 2, subfamily B, polypeptide 6 (CYP2B6) or cytochrome P450, family 2, subfamily C, polypeptide 9 (CYP2C9) that have narrow therapeutic index, and that cannot be discontinued before starting treatment with LDE225
* Patients unwilling or unable to comply with the protocol
* Requirement for anticoagulation with warfarin