Discontinuity at the transition between the inflationary and radiation-dominated universe

Inflation is thought to be an extraordinarily brief period during the very early universe when the scale factor increased a remarkable $e^{60}$-fold, however this also comes with a corresponding increase in the expansion rate $dot{a}=mathrm{d}a/mathrm{d}t$. When inflation ends, the exponential acceleration ceases and we transition to the radiation-dominated era where the expansion rate now evolves as $dot{a}_text{rad}(t)propto t^{-1/2}$.

But what happened to the expansion rate, $dot{a}_text{end}$, at the end of inflation? Did it become an initial condition of the FLRW universe? Surely it’s much too large to have decreased under $ddot{a}_text{rad}(t)$ in any reasonable amount of time, before the universe become too diluted? Did it undergo a large deceleration, $ddot{a}(t)ll 0$, a sort of "anti-inflation" right at the end of the inflation period? If so, what is the reason for this deceleration?

To make this more exact, consider inflation to begin and end at $t=0$ and $t=T$ respectively, and assume that inflation expands the universe such that $a(T)/a(0)=e^{60}$. We can then describe the inflation as

begin{align}
a(t<T)=a(0)exp(ht)=a(T)expleft(60left(frac{t}{T}-1right)right)
end{align}
where $h$ is the (constant) Hubble parameter during inflation. For $t>T$ we transition to the radiation-dominated universe and so to maintain continuity of the scale factor we must have
begin{align}
a(t>T)=a(T)sqrt{t/T}
end{align}
So that’s all well and good. Now let’s consider $dot{a}$. Taking the time derivatives of each of the above gives
begin{align}
dot{a}(t<T)=frac{60}{T}a(T)expleft(60left(frac{t}{T}-1right)right)=frac{60}{T}a(t<T)
end{align}
and
begin{align}
dot{a}(t>T)=frac{a(T)}{2T}frac{1}{sqrt{t/T}}
end{align}
Now we find that the continuity is broken at $t=T$, with the former giving $dot{a}(T)=60 a(T)/T$ and the latter giving $dot{a}(T)=a(T)/2T$, which is not surprising since we are modelling this as an instantaneous transition between inflation and the radiation-doimated universe. So assuming this is valid we see that the expansion rate has decreased by a factor of 120, which is almost completely negligible when compared to the $e^{60}$-fold increase during inflation. So this can’t be the decrease we desire.

What gives? Am I missing something crucial? How did we go from this ludicrously fast expansion to what we have today?

Additional question: assuming my math is even valid, is there a reason for the 120-fold decrease in the expansion rate? If we go by the standard "slow roll" explanation for a scalar field $phi$ then inflation occurs when $V(phi)ggdot{phi}^2$ and stops when the potential decreases enough so that $V(phi)approxdotphi^2$, is this what causes the large negative acceleration required? If so is it able to be shown explicitly?