Stopping inflation, dark energy and switching it on again Inflation is the current paradigm to explain several big problems of current cosmological models. 
My question is simple. Why or what stopped inflation and why is the cosmic acceleration behaviour just like a "little inflation"? Can we guess good reasons of why inflation era terminated and originated the current Universe and now accelerating the universe? That is, could we consider the current cosmic acceleration caused by an inflation-like scalar field? How or what could we build a scalar potential that:
1) Be able to explain the past inflationary epoch.
2) Be able to fit the current cosmic positively accelerated Universe.
Is 1)->2) Necessary or could be independent?
Remark: I am aware that there are lots of inflationary models waiting to be tested the next years to come, but I wondered why or how did inflation to stop because in an almost de Sitter or dS in the asymptotic future we could (or not!) stop the Big Rip menace...
 A: at least in the case of the (favoured) slow-roll scenarios, inflation stops rather automatically at a certain point. 
How do slow-roll models work? One assumes that there is a scalar field $\phi$ with potential $V(\phi)$, which is minimally coupled to gravity, i.e. 
\begin{equation}
S \supset \int d^4x \sqrt{-g}\left( \frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi \nabla_{\nu}\phi - V(\phi) \right).
\end{equation}
The logic of slow-roll inflation is that the inflaton field $\phi$ rolls down e.g. a rather flat potential, i.e. while $\phi$ keeps rolling the potential energy of the field is roughly constant, so it almost has the effect of a cosmological constant. But only almost, because $\phi$ will at a certain point reach a rather steep region of $V$ and inflation can stop. 
This may help to get an intuition. 
More mathematically, one can define slow-roll parameters $\epsilon$ and $\eta$ as follows: 
\begin{equation}
\epsilon = \frac{1}{2} \left(\frac{V^{\prime}}{V}\right)^2
\end{equation}
and 
\begin{equation}
\eta = \frac{V^{\prime \prime}}{V}
\end{equation}
where the $\prime$ denote derivatives with respect to $\phi$. It can be shown that the conditions 
\begin{equation}
\epsilon \ll 1 ~~~ \text{and} ~~ |\eta| \ll 1
\end{equation}
are sufficient to obtain inflation (in the sense of sufficiently rapid expansion). 
If one of those two conditions fails to hold, e.g. because the potential is somewhere too steep or the curvature is too large, then slow-roll inflation ends. 
In fact, making inflation stop is not so hard for model builders. The more complicated problem is to find a system with flat potential that remains flat after accounting for quantum corrections (and possibly other stuff).
Thus, your question essentially boils down to how does $V$ look like. And that we don't know. 
I hope that this is helpful! 
Recommendations for reading: 
TASI lectures on inflation by Daniel Baumann
psm
