Apparently, the paper "ON THE PROBABILITY THAT A RANDOM ±1-MATRIX IS SINGULAR" by Kahn, Komlos and Szemeredi (Corollary 4 therein) answers my question, and states that it is $(1+o(1))2\binom{m}{2}/2^n$, thereby improving the bound from exponential in $m$ to polynomial in $m$.

https://www.ams.org/journals/jams/1995-08-01/S0894-0347-1995-1260107-2/S0894-0347-1995-1260107-2.pdf