= Solution
For the radiation-era law beginning at $t_r$, the present particle-horizon estimate is
$$
d_H(t_0)=2ct_0\left(1-\sqrt{t_r/t_0}\right)\simeq2ct_0\simeq6\times10^{25}\ \mathrm m.
$$
Rescale this present physical distance back to $t_r$ using $a(t_r)/a(t_0)=\sqrt{t_r/t_0}$. Its physical size was $2c\sqrt{t_rt_0}\simeq0.6\ \mathrm m$, of order one metre as required. The matched inflationary Hubble rate is $H=1/(2t_r)\simeq5\times10^{34}\ \mathrm{s}^{-1}$, with Hubble radius $c/H\simeq6\times10^{-27}\ \mathrm m$.
If inflation produces $N$ e-folds before $t_r$, this patch began with size $0.6e^{-N}$ metres. It fits within one Hubble radius if
$$
\boxed{e^N\gtrsim10^{26},\qquad N\gtrsim26\log10\simeq60.}
$$
This estimate uses the idealized radiation-only history stated in the question; substituting the actual matter-dominated history would change the numerical estimate.
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