We present the detection of multiple carbon monoxide CO line transitions with ALMA in a few tens of infrared-selected galaxies on and above the main sequence at z = 1.1-1.7. We reliably detected the emission of CO (5 -4), CO (2 -1), and CO (7 -6)+[C I](3P2-3P1) in 50, 33, and 13 galaxies, respectively, and we complemented this information with available CO (4 -3) and [C I](3P1-3P0) fluxes for part of the sample, and by modeling of the optical-to-millimeter spectral energy distribution. We retrieve a quasi-linear relation between LIR and CO (5 -4) or CO (7 -6) for main-sequence galaxies and starbursts, corroborating the hypothesis that these transitions can be used as star formation rate (SFR) tracers. We find the CO excitation to steadily increase as a function of the star formation efficiency, the mean intensity of the radiation field warming the dust (〈 U〉), the surface density of SFR (ςSFR), and, less distinctly, with the distance from the main sequence (ΔMS). This adds to the tentative evidence for higher excitation of the CO+[C I] spectral line energy distribution (SLED) of starburst galaxies relative to that for main-sequence objects, where the dust opacities play a minor role in shaping the high-J CO transitions in our sample. However, the distinction between the average SLED of upper main-sequence and starburst galaxies is blurred, driven by a wide variety of intrinsic shapes. Large velocity gradient radiative transfer modeling demonstrates the existence of a highly excited component that elevates the CO SLED of high-redshift main-sequence and starbursting galaxies above the typical values observed in the disk of the Milky Way. This excited component is dense and it encloses ∼50% of the total molecular gas mass in main-sequence objects. We interpret the observed trends involving the CO excitation as to be mainly determined by a combination of large SFRs and compact sizes, as a large ςSFR is naturally connected with enhanced dense molecular gas fractions and higher dust and gas temperatures, due to increasing ultraviolet radiation fields, cosmic ray rates, as well as dust and gas coupling. We release the full data compilation and the ancillary information to the community.