If you do not anticipate any significant increase in enrollment, you may want to consider consolodating with the other PTO's at some point. But for this year, give the secretary a shot.
I think it would be difficult to manage every year with only 2 willing parents. And you may find that the other groups are fighting a similar battle, but by banding together, you could pool everyone's talent and resources.
Answer is the same, either way - it depends on if your bylaws prevent it.
But these could be two different issues:
1) Must a member/officer be a parent of a current student?
2) Can a school employee be a PTO officer? (if it's true she is the school secretary)
#1 is routinely a requirement in PTO bylaws. But not always. Some don't address the issue and some take the position anyone in the community can be a member. (Note - you need to look both at who can be a MEMBER and who can be an OFFICER in your bylaws.)
#2 varies. I've been a member of PTOs that took opposite views. One prohbited school staff members from serving as an officer, although they could be members and committee chairs. The other left it open and we had the school nurse as president a couple of years. (She was also a parent.)
With your school being so small, you probably need to keep your options open as much as possible.
If she's the school secretary, at least you know she'll be around and involved and has a vested interest. If she's merely the PTO secretary, I'd wonder if her interest would wane since her child is no longer there.
Robert's Rules is silent on this issue as it is not parliamentary in nature. The answer will be found in your by-laws, which no longer require student attendance at your school as a prerequisite for a person to run for and hold a position on the PTO.
My opinion is that you ask the secretary to hold off assuming the President's role until late August. Broadcast the board position openings loud and wide through-out the summer and early fall. If it appears to the parents that the PTO is in real danger of folding without more volunteer help, it's possible someone will step up to fill the President's role in the end.
If that doesn't happen, the secretary can be given the spot in a "Hail Mary" move.
We have a VERY small school of 73 kids. It is extremely hard to find volunteers. We have only been around for 3 years and this year was extremely successful. There are only 4 parents on the PTO who do all of the work, and two of them are board members who are leaving office. Our Secretary is interested in being President to keep it from folding, but her child is going to another school in the fall. Is this allowed? My husband says it may be breaking federal or state laws, but I thought we govern ourselves? We are in the process of becoming a corporation and filing for 501c3. Our original bylaws specifically stated a member must have a child in the school, but that was removed when we revised it last year. Our school program is one of a bunch of satellites that branch off from a main school. Some of these satellites have their own PTO's. Her son is attending the main school, so technically, he is still a student. Anyone with experience in this matter?