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Recently, robust $d$-wave superconductive (SC) order has been unveiled in the ground state of the 2D $t$-$t'$-$J$ model -- with both nearest-neighbor ($t$) and next-nearest-neighbor ($t'$) hoppings -- by density matrix renormalization group studies. However, there is currently a debate on whether the $d$-wave SC holds up strong on both $t'/t>0$ and $t'/t<0$ cases for the $t$-$t'$-$J$ 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 $t$-$t'$-$J$ model on cylinders with widths up to $W=6$ and down to low temperature as $T/J \simeq 0.06$, pushing the boundaries of contemporary finite-$T$ calculations. For $t'/t>0$, we find a dome-like SC regime with a diverging $d$-wave pairing susceptibility, $χ_\textrm{SC} \propto 1/T^α$ below a characteristic temperature $T_c^*$. Near optimal doping, $T_c^*$ reaches its highest value of about $0.15 J$. Above $T_c^*$ yet below a higher crossover temperature $T^*$, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for $t'/t<0$ 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 $t$-$t'$-$J$ 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.