Phase II Study of Immunomaintenance Using POmalidomide With Elotuzumab afteR Second Autologous Transplant

Inclusion Criteria: * Patients with a diagnosis of relapsed or relapsed/refractory multiple myeloma who will be undergoing a second unplanned autologous peripheral blood stem cell transplant for their disease. * Age ≥ 18 years at the time of consent. * Written informed consent and HIPAA authorization for release of personal health information. * NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * ECOG Performance Status of 0, 1, or 2 within 14 days prior to registration. * Patients must have had measurable disease, defined as one of the following: * monoclonal protein (M-protein): ≥ 0.5g/dL on serum protein electrophoresis or ≥ 200 mg of monoclonal protein on a 24-hour urine protein or involved serum light chain ≥ 10 mg/dl at time of relapse, leading to decision to proceed to transplant; or * biopsy proven plasmacytoma that can be assessed by physical exam or imaging; or * if non-secretory, ≥10% plasma cells on BM biopsy/aspirate at time of relapse or plasmacytoma as described above. * NOTE: Urine protein electrophoresis (UPEP) (on a 24-h collection) is required, no substitute method is acceptable. Urine must be assessed to establish response if the baseline urine M-spike is ≥ 200 mg/24 h at the time of enrollment. Please note that if both serum and urine M-components are present prior to transplant, both should be assessed in order to evaluate response. * Patients may have received any number and type of previous treatments for myeloma including elotuzumab or pomalidomide, but not simultaneous administration of these two agents. * Previous allogeneic transplant is allowed provided the patient is not receiving ongoing therapy for GVHD. * Previous CAR-T transplantation or other BMCA directed therapy is also allowed provided there is no evidence of residual cytokine release syndrome or cytokine release encephalopathy syndrome. * As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study. * Pomalidomide will not be provided the study and therefore study subjects must have confirmed access to pomalidomide for use during the study established at time of enrollment. Furthermore, subjects must meet all of the following applicable inclusion criteria 45-90 days post-transplant to be treated on this study: * Patients must have recovered from transplantation to ≤grade 2 non- hematologic toxicity, with the exception of alopecia. * No evidence of progression of myeloma noted within 45 days post-transplant. * Demonstrate adequate organ function as defined in the protocol; all screening labs to be obtained within 14 days prior to initiation of treatment. * Females of childbearing potential must have two negative pregnancy tests (serum or urine): within 14 days and 24 hours prior to treatment. \-- NOTE: Females are considered of childbearing potential unless they are surgically sterile (have undergone a hysterectomy, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months. * Females of childbearing potential and males must be willing to abstain from heterosexual activity or to use 2 forms of effective methods of contraception from the time of informed consent until 5 months for females, and 7 months for males after treatment discontinuation. The two contraception methods can be comprised of two barrier methods, or a barrier method plus a hormonal method. Interventions such as IUD, tubal ligation, hormonal (birth control pills, injections, hormonal patches, vaginal rings, or implants), or partner's vasectomy, all count as one method. For WOCBP, a second form must also be used. * As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study. * Subjects must be willing to provide BM, stool and blood samples during the study period. Exclusion Criteria: * Active infection requiring systemic therapy. * Known additional malignancy that is active and/or progressive requiring treatment; exceptions include basal cell or squamous cell skin cancer, in situ cervical or bladder cancer, or other cancer for which the subject has been disease-free for at least three years. Patients who have undergone a curative procedure for another malignancy are eligible. * Active central nervous system (CNS) metastases. * Treatment with any investigational drug within 30 days prior to registration. * Planned transplant is considered part of tandem approach for newly diagnosed MM * History of severe hypersensitivity reaction to any monoclonal antibody. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because pomalidomide is an agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with pomalidomide, breastfeeding should be discontinued if the mother is treated with pomalidomide. * Patients with known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) might be enrolled if the viral load by polymerase chain reaction (PCR) is undetectable with/without active treatment and absolute lymphocyte count \>= 350/ul. Such subjects may stay on antiviral therapy during study treatment. * Patients with a positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection might be enrolled if the viral load by PCR is undetectable with/without active treatment. Such patients may stay on viral therapy while on treatment.