Adding the Immunotherapy Drug Cemiplimab to Usual Treatment for People With Advanced Non-Small Cell Lung Cancer Who Had Previous Treatment With Platinum Chemotherapy and Immunotherapy (An Expanded Lung-MAP Treatment Trial)

Inclusion Criteria: * Participants must have been assigned to S1800E by the Southwest Oncology Group (SWOG) Statistics and Data Management Center (SDMC). Assignment to S1800E is determined by the LUNGMAP protocol * Participants must have measurable or non-measurable disease documented by CT or MRI. The CT from a combined positron emission tomography (PET)/CT may be used to document only non-measurable disease unless it is of diagnostic quality. Measurable disease must be assessed within 28 days prior to randomization. Pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease. Non-measurable disease must be assessed within 42 days prior to randomization. All disease must be assessed and documented on the Baseline Tumor Assessment Form. Participants whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to registration * Participants must have a CT or MRI scan of the brain to evaluate for central nervous system (CNS) disease within 42 days prior to randomization * Participants must have received exactly one anti-PD-1 or anti-PD-L1 therapy for advanced disease (stage IV or recurrent disease, or stage I-III disease in certain circumstances outlined below). Anti-PD-1 or anti-PD-L1 therapy may have been given alone or in combination with other therapy. For participants who received neoadjuvant, adjuvant, and/or consolidation anti-PD-1 or anti-PD-L1 therapy for stage I-III disease: * If they experienced disease progression within (≤) 365 days from initiation (cycle 1 day 1) or anti-PD-1 or anti-PD-L1 therapy, this counts as the single allowed anti-PD-1 or anti-PD-L1 therapy for advanced disease * If they experienced disease progression more than (\>) 365 days from initiation (cycle 1 day 1) or anti-PD-1 or anti-PD-L1 therapy, this is not considered anti-PD-1 or anti-PD-L1 therapy for advance disease. These participants must have received anti-PD-1 or anti-PD-L1 therapy for stage IV or recurrent disease * Participants must have experienced disease progression (in the opinion of the treating investigator) more than (\>) 84 days following initiation (cycle 1 day 1) of their most recent anti-PD-1 or anti-PD-L1 therapy * Participants who received anti-PD-1 or anti-PD-L1 therapy for stage IV or recurrent disease must have had a best response of stable disease, partial response or complete response (in the opinion of the treating investigator) on the anti-PD-1 or anti-PD-L1 therapy for stage IV or recurrent disease * Participants must have received platinum-based chemotherapy and experienced disease progression (in the opinion of the treating investigator) during or after this regimen * Participants with a known sensitizing molecular alteration for which a Food and Drug Administration (FDA)-approved targeted therapy for NSCLC exists (e.g., EGFR, ALK, ROS1, BRAF, RET, NTRK, KRAS, HER2, and MET sensitizing mutations), must have previously received at least one of the approved therapy(s). Prior targeted therapy for participants with targetable alterations is allowed if all other eligibility criteria are also met * Participants must have recovered (≤ grade 1) from any side effects from the most recent anti-cancer treatment prior to randomization * Participants must not have received prior therapy with docetaxel for this disease * Participants must not have received any palliative radiation therapy within 14 days (or palliative bone radiation therapy within 7 days) prior to randomization * Participants must not be planning to receive any concurrent chemotherapy, immunotherapy, or biologic therapy for cancer treatment while receiving treatment on this study * Participants must not have undergone major surgery within 28 days prior to randomization, or subcutaneous venous access device placement within 7 days prior to randomization. Participants must not have postoperative bleeding complications or wound complications from a surgical procedure performed within 2 months prior to randomization. The participant must not have elective or planned major surgery to be performed during the course of this study * Absolute neutrophil count ≥ 1.5 x 10\^3/uL (within 28 days prior to randomization) * Hemoglobin ≥ 9.0 g/dL (within 28 days prior to randomization) * Platelets ≥ 100 x 10\^3/uL (within 28 days prior to randomization) * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) (within 28 days prior to randomization) unless history of Gilbert's disease. Participants with history of Gilbert's disease must have total bilirubin ≤ 5 x institutional ULN * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 x institutional ULN (within 28 days prior to randomization). Participants with history of liver metastasis must have AST and ALT ≤ 5 x ULN * Participants must have a creatinine ≤ the institutional (I)ULN or calculated creatinine clearance ≥ 50 mL/min using the following Cockcroft-Gault formula. This specimen must have been drawn and processed within 28 days prior to randomization * Participants must have a urinary protein test performed within 28 days prior to randomization * Participants' most recent Zubrod/Eastern Cooperative Oncology Group (ECOG) performance status must be 0-1 and be documented within 28 days prior to randomization * Participants must have a completed medical history and physical exam within 28 days prior to randomization * Participants with evidence of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load while on suppressive therapy on the most recent test results obtained within 6 months prior to randomization, if indicated by the treating investigator * Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. Participants currently being treated for HCV infection must have undetectable HCV viral load test on the most recent test res