Recently, robust -wave superconductive (SC) order has been unveiled in the ground state of the 2D model—with both nearest-neighbor () and next-nearest-neighbor () hoppings—by density matrix renormalization group studies. However, there is currently a debate on whether the -wave SC holds up strong on both and cases for the model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the model on cylinders with widths up to and down to low temperature as , pushing the boundaries of contemporary finite- calculations. For , we find a domelike SC regime with a diverging -wave pairing susceptibility, below a characteristic temperature . Near optimal doping, reaches its highest value of about . Above yet below a higher crossover temperature , the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for , we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.