I contacted AA. Found out that you can qualify with previously-purchased travel, AND each segment in paid F is considered a separate flight.
In my case, I have 2-segment paid F flights on April 14 and June 8. So apparently I will get the bonus for those flights, plus any others taken from June 9-July 8. Don't know that I'd have gone out of my way to get it, but it's a nice perk, and if I have to take a trip in July, I might try to do it earlier in the month rather than later.