If I understand the question correctly, brute-force search with some Julia code gives following small values

\begin{array}{rr|rrrrrrrrrrrrrrrrr}
&& n\\
&& 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14\\
\hline
m & 2 & 4 & = \\
  & 3 &  & 7 & 8 & = \\
  & 4 &  &   & 11 & 13 & 14 & 15 & 16 & = \\
  & 5 &  &   &    & 16 & 19 & 21 & 23 & 25 & 27 & 30 & 31 & 32 & = \\
  & 6 &  &   &    &    & 22 & 26 & 29 & 32 & 35 & 38 & 41 & 45 & 48 \\
  & 7 &  &   &    &    &    & 29 & 34 & 38 & 42 & 46 & 50 & 54 & 58 \\
  & 8 &  &   &    &    &    &    & 37 & 43 & 48 & 53 & 58 & 63 & 68 \\
  & 9 &  &   &    &    &	&    &    & 46 & 53 & 59 & 65 & 71 & 77 \\
  & 10&  &   &    &    &    &    &    &    & 56 & 64 & 71 & 78 & 85 \\
  & 11&  &   &    &    &    &    &    &    &    & 67 & 76 & 84 & 92 \\
  & 12&  &   &    &    &    &    &    &    &    &    & 79 & 89 & 98 \\
  & 13&  &   &    &    &    &    &    &    &    &    &    & 92 & 103 \\
  & 14&  &   &    &    &    &    &    &    &    &    &    &    & 106 \\
\end{array}

"=" means that the rest of the row repeats the last value, that is, there is a string that exhibits all $2^m$ subsets (adding more length will not change this).

Looking at diagonals:
 * The main diagonal is as described in the question.
 * The first superdiagonal 4,8,13,19,26,34,43,53,64,76 has [one OEIS match][1], A034856 "a(n) = binomial(n+1, 2) + n - 1 = n*(n + 3)/2 - 1".
 * The second superdiagonal has no OEIS entry but OEIS [guesses][2] it to be a quadratic.  Same with third superdiagonal.
 * The fourth superdiagonal, starting from 16,25,35,46,58,71,85,100,116,133,151 has [two matches][3] (but different values before the 16).

If we write $T(m)$ for the smallest $n$ where $2^m$ appears, $T$ forms a sequence that starts 1,2,4,8,13, which matches many things in OEIS. Examples of shortest strings where all $2^m$ subsets of alphabet $\{1,2,\ldots,m\}$ appear are:

\begin{array}{l|l}
m & T(m) \\
\hline
1 & 1 & 1 \\
2 & 2 & 12 \\
3 & 4 & 1231 \\
4 & 8 & 12314234 \\
5 & 13 & 1234512413524 \\
\end{array}


  [1]: https://oeis.org/search?q=4%2C8%2C13%2C19%2C26%2C34%2C43%2C53%2C64%2C76&sort=&language=&go=Search
  [2]: https://oeis.org/search?q=8%2C14%2C21%2C29%2C38%2C48%2C59%2C71%2C84&sort=&language=&go=Search
  [3]: https://oeis.org/search?q=16%2C25%2C35%2C46%2C58%2C71%2C85%2C100%2C116%2C133%2C151&sort=&language=&go=Search