We present the discovery and characterization of five hot and warm Jupiters—TOI-628 b (TIC 281408474; HD 288842), TOI-640 b (TIC 147977348), TOI-1333 b (TIC 395171208, BD+47 3521A), TOI-1478 b (TIC 409794137), and TOI-1601 b (TIC 139375960)—based on data from NASA's Transiting Exoplanet Survey Satellite (TESS). The five planets were identified from the full-frame images and were confirmed through a series of photometric and spectroscopic follow-up observations by the TESS Follow-up Observing Program Working Group. The planets are all Jovian size (RP = 1.01-1.77 RJ) and have masses that range from 0.85 to 6.33 MJ. The host stars of these systems have F and G spectral types (5595 ≤ Teff ≤ 6460 K) and are all relatively bright (9.5 < V < 10.8, 8.2 < K < 9.3), making them well suited for future detailed characterization efforts. Three of the systems in our sample (TOI-640 b, TOI-1333 b, and TOI-1601 b) orbit subgiant host stars ( $\mathrm{log}$ g < 4.1). TOI-640 b is one of only three known hot Jupiters to have a highly inflated radius (RP > 1.7 RJ, possibly a result of its host star's evolution) and resides on an orbit with a period longer than 5 days. TOI-628 b is the most massive, hot Jupiter discovered to date by TESS with a measured mass of ${6.31}_{-0.30}^{+0.28}$ MJ and a statistically significant, nonzero orbital eccentricity of e = ${0.074}_{-0.022}^{+0.021}$ . This planet would not have had enough time to circularize through tidal forces from our analysis, suggesting that it might be remnant eccentricity from its migration. The longest-period planet in this sample, TOI-1478 b (P = 10.18 days), is a warm Jupiter in a circular orbit around a near-solar analog. NASA's TESS mission is continuing to increase the sample of well-characterized hot and warm Jupiters, complementing its primary mission goals.