The big bang (a singularity 13.8 billion years ago that was the beginning of time) didn't happen at all. Inflationary cosmology consists of:
ΛCDM (aka "big bang") cosmology starting at a cosmological time of some small fraction of a second, preceded by
an inflationary epoch lasting for an unknown amount of time (but at least 50-60 e-folds), preceded by
there may be no way to know, because inflation erases any hints of what came before.
Times quoted as "after the big bang" are ΛCDM model times and don't count from the true beginning of time. There's a common misconception that if inflation ended $10^{-32}$ seconds "after the big bang", that it must have lasted for no longer than $10^{-32}$ seconds. In reality, it has to last longer than that in order to accomplish its goal of producing smooth starting conditions for the ΛCDM epoch, and in most (all?) specific models it lasts for far longer than the minimum time. There are eternal inflation models in which it lasts forever.
We don't know how old the universe is. We only know that the current era of (non-inflationary) expansion started about 13.8 billion years ago.