Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes.  When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out.  After all, nearly all the members of the volunteer group have other jobs.

Hannah comes up with a plan that the committee likes.  Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month.  Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee $\textit{must have}$ taught together twice (all five members are the same during two different months)?
1. **Calculate the total number of possible committees without restrictions:**
   The total number of ways to choose 5 members out of 13 is given by the binomial coefficient:
   \[
   \binom{13}{5} = \frac{13!}{5!(13-5)!} = \frac{13!}{5! \cdot 8!} = 1287
   \]

2. **Calculate the number of committees consisting only of women:**
   There are 6 women in the group. The number of ways to choose 5 women out of 6 is:
   \[
   \binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5! \cdot 1!} = 6
   \]

3. **Calculate the number of committees consisting only of men:**
   There are 7 men in the group. The number of ways to choose 5 men out of 7 is:
   \[
   \binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} = 21
   \]

4. **Calculate the number of valid committees (at least one woman and one man):**
   The total number of committees without restrictions is 1287. We subtract the number of all-women and all-men committees:
   \[
   1287 - 6 - 21 = 1260
   \]

5. **Apply the Pigeonhole Principle:**
   According to the Pigeonhole Principle, if we have more committees than the number of possible unique committees, at least one committee must repeat. Therefore, to ensure that at least one committee has taught together twice, we need:
   \[
   1260 + 1 = 1261 \text{ months}
   \]

The final answer is $\boxed{1261}$