Did the rapid inflationary expansion slow down a lot because the inflaton field had decayed, or because of gravity from matter/radiation? What happened to the rate of expansion right after inflation ended?
Did the rapid inflationary expansion slow down a lot because the inflaton field had decayed, or because gravity from matter/radiation gradually decelerated the rapid expansion? 
In other words, did inflationary expansion instantaneously turn into non-inflationary because inflaton field had decayed, or did inflationary expansion gradually decelerate to be non-inflationary because of gravity from matter/radiation?
 A: The inflationary expansion gradually decelerated. The change was not instantaneous, but “gradually” means over something like a billionth of a trillionth of a trillionth of a second!
In most inflationary models, the value of a scalar field filling all of space transitions from a nonzero value to a zero value as the field “rolls down its potential energy hill” to the bottom. The potential energy here is the energy of the field interacting with itself.
For example, a simple and classic model (but one no longer used in inflationary models) is the “Mexican hat” potential, $$V(\phi)=a(|\phi|^2-b^2)^2.$$ At high temperature, the field has the value 0 and the nonzero potential energy drives inflation. As the field evolves toward a value with $|\phi|=b$, the potential energy gradually drops to zero and inflation gradually stops.
Update to new questions posed by OP on 1/29:

Did the rapid inflationary expansion slow down a lot because the inflaton field had decayed, or because gravity from matter/radiation gradually decelerated the rapid expansion?

Inflation slows down and stops because the inflaton field decays. 

In other words, did inflationary expansion instantaneously turn into non-inflationary because inflaton field had decayed, or did inflationary expansion gradually decelerate to be non-inflationary because of gravity from matter/radiation?

Neither. Inflation gradually turns into non-inflationary expansion because the inflaton field decays. Nothing happens instantaneously in this process. 
A: 
Actually my question is about the rate of expansion right after
inflation ended when the field reach the bottom of the potential.

For the rate of expansion (the Hubble parameter) just take c over the Hubble radius:

So the Hubble parameter right after inflation in SI units was about 
$$ \rm H = \frac{c}{r_H} = \frac{3\cdot 10^8 \ m/s}{10^{-28.5} \ m} = \frac{10^{37}}{s}$$
That number might vary depending on the specific model, but at least the order of magnitude is in that range.
A: No, inflationary expansion did not “instantly turn into non-inflationary”. This epoch ended when the huge potential energy of the inflaton field started to decay into matter particles and radiation, a process which is called reheating. Before reheating the universe expanded exponentially and thereafter decelerated. Most physicists agree on this model. As far as I can tell the era where reheating happened was very short but certainly not zero.
