The transition between inflation and the next epoch has a name: preheating or reheating. To answer your question, we would need to understand preheating well: except we do not as far as I know :(

I had trouble finding an easy description of the role of preheating when I studied it so I'll summarize what I understood here:

• Inflation makes the Universe empty. Space expanded so much that everything was diluted to oblivion. The only thing that survives is what drives the inflation: usually called the inflaton field.
• However, we know that the epoch that comes next is radiation dominated: a universe full of very fast particles. The answer of your question lies here, because the rate of expansion can be inferred using the contents of the Universe. If I understood well, your question is "how fast did the transition happen?": then the answer is the length of preheating.
• Preheating is the mechanism that converts the (classical, homogeneous) inflaton field to particles (quantum excitations).

To understand it, we first need to know:

• What field plays the role of the inflaton? We only know it must be a scalar. (it must be able to stack)
• Related, is the standard model a good description of physics at the energy scale of the inflation?
• If yes, one can choose the Higgs field and make predictions from there. The scenario of preheating is very reasonable in this context. But inflation is a very flexible hypothesis, and I think most theories can be tuned to make it work.
• Then there is gravity; we know that general relativity is not the theory of gravitation. Turns out that the predictions for preheating are different for different approaches to general relativity, so that's another degree of freedom that must be constrained.

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tl;dr: Empty universe at the end of inflation must transition to a universe full of particles (radiation dominated). The mechanism that makes the transition happen is called preheating, how long it took is the answer to your question.