  Over the last few years, a large family of cosmological attractor models has
been discovered, which can successfully match the latest inflation-related
observational data. Many of these models can also describe a small cosmological
constant $\Lambda$, which provides the most natural description of the present
stage of the cosmological acceleration. In this paper, we study
$\alpha$-attractor models with dynamical dark energy, including the
cosmological constant $\Lambda$ as a free parameter. Predominantly, the models
with $\Lambda > 0$ converge to the asymptotic regime with the equation of state
$w=-1$. However, there are some models with $w\neq -1$, which are compatible
with the current observations. In the simplest models with $\Lambda = 0$, one
has the tensor to scalar ratio $r=\frac{12\alpha}{N^2}$ and the asymptotic
equation of state $w=-1+\frac{2}{9\alpha}$ (which in general differs from its
present value). For example, in the seven disk M-theory related model with
$\alpha = 7/3$ one finds $r \sim 10^{-2}$ and the asymptotic equation of state
is $w \sim -0.9$. Future observations, including large-scale structure surveys
as well as B-mode detectors will test these, as well as more general models
presented here. We also discuss gravitational reheating in models of
quintessential inflation and argue that its investigation may be interesting
from the point of view of inflationary cosmology. Such models require a much
greater number of $e$-folds, and therefore predict a spectral index $n_{s}$
that can exceed the value in more conventional models by about $0.006$. This
suggests a way to distinguish the conventional inflationary models from the
models of quintessential inflation, even if they predict $w = -1$.
