Hmmm. Well, I hope you don’t mind me jumping in here, but since no-one else has answered you I’ll give it a shot. I work for a Government organisation in Australia with very strict rules about how we are allowed to recruit, so your experience is likely to be very different. I suppose my advice is worth what you paid for it. 
Having sat on a fair number of interview panels in the last few years, I do have to say that the most likely reason we’d readvertise a position was because the candiates we had weren’t what we were looking for. In order to save everybody’s time the new ad would say something like, “previous candiates need not reapply”.
Rarely a position will be withdrawn and readvertised because of some internal politicing about the position itself, in which case there’s be no reason why previous candiates couldn’t reapply.
However, if someone on the panel is happy for you to reapply, and you’re sure they’re not just being kind, there are some things you could do to improve your chances next time.
If you think you missed out because you didn’t interview well, then you’re quite entitled to go back to the panel and ask for some feedback on your performance. We have to take copious notes about each candidate because they have a right to come back to the panel chair and ask for feedback.
Maybe ask the person who you’ve been talking to already about what you could do better in an interview. Were there questions you answered badly? Were there things they were looking for that you didn’t provide? Try and get some idea of what you need to improve upon.
Can’t hurt to ask them what they’re looking for in a candidate, they’ll probably answer in general terms, but you’ll get a few clues.
Bear in mind that panels will ask the numbers of people the same questions over and over and it gets very tiring. If you can answer questions intelligently, succinctly and without excessive waffle the panel is likely to remember you with gratitude.
Possibly your comment about lack of managerial experience in previous directors is a clue…volunteer organisations need really good management skills because they’re working with such a wide range of different people with different committments and reasons for being there.
One thing which is worth considering is how you positioned yourself with regards to the organisation during the interview. I’ve struck this a lot with people applying for jobs which are taking them from project work up to the next level of management. A significant number of candiates haven’t made the intellectual leap to thinking in terms of big picture/policy/stategic directions which managment requires, and tend to answer questions from the “shop floor” perspective.
When we’re interviewing for managment-type positions we’re really looking for people who’ve started thinking in terms of policy and direction rather than individual projects or jobs.
If you think this might be an issue with you, you could go back and rethink the questions in terms of how you would answer them from a management perspective. Imagine you were the director of this organisation, what would be the issues you’d have to deal with? How would you manage it? What things need to happen/be changed/improved/removed?
Whether they ask the same questions or different ones next time, the questions will cover pretty much the same ground, so you can start thinking about your replies before the interview. You might expect, for example, a question which asks about your experience. Think about an answer which illustrates you in the role of management, even if you have to stretch a bit.
If there are selection criteria for the position even better, you have a big clue about what the panel is looking for, and you get the opportunity to present those parts of your experience which answer their needs.
You might want to revisit your resume, make sure that it reflects the kinds of skills you’re going to need as director of this organisation.
If you get a second interview treat it as a whole new deal. You might say something such as “thanks for interviewing me again”, but otherwise treat the second interview as the main event. You didn’t get the job last time, so referring to the previous interview isn’t likely to be helpful.
Sorry this has been so long, I’m prolix at the best of times.
Good luck!
PS Hello SDMB…I’ve been lurking for a long time, but this is first time I’ve had something to say which hasn’t already been said, and better. I think I have the hang of the rules, but I beg your indulgence if I’ve managed to get it wrong.