The primordial universe is best described by the theory of cosmological inflation which predicts an early epoch of quasi-exponential expansion. Till date, observations of the Cosmic Microwave Background (CMB) have successfully constrained several models of inflation by comparing their predictions with the observed primordial scalar power spectrum with amplitude $P_{\zeta}(k) ~10^{-9}$ and a red-tilt. However, CMB surveys (e.g. Planck) can only probe ~ 10-15 e-folds of inflation near the CMB pivot scale $k=0.002 Mpc^{-1}$ and therefore fluctuations at smaller scales of inflation can be large enough to lead to large density contrast, which can collapse in the post-inflationary epochs to form primordial black holes (PBH). For PBH to contribute a considerable fraction of cold dark matter, the required amplitude of primordial scalar perturbations is quite large ($P_{\zeta}(k) ~10^{-2}$ ) if they are formed in radiation epoch. In alternate cosmological histories, where additional epochs of arbitrary equation of state precede radiation epoch, the dynamics of PBH formation and relevant mass ranges can be different leading to requirement of lower amplitude of primordial power at smaller scales of infl ation. This alternate history can also modify the predictions for the gravitational wave (GW) spectrum, which can be probed by upcoming GW observations. Specifically, an early kination epoch, which is prevalent in quintessential inflation models, can lead to percent level abundance of PBH for a lower amplitude of $P_{\zeta}(k)$ as compared to PBH formation in a standard radiation epoch. Moreover, such an early kination epoch affects the evolution of first and second order GW spectrum predicting enhancement in the amplitude of the GW spectrum in a kination epoch with respect to that in a standard radiation epoch.