Last year, we played an initial doubleheader right after VHSL regionals. We still had a handful of guys alive in the state tourney, so we had to have a few fill-in players that day. But it worked out.
By the time we got to the following weekend, I think we had only 1-2 guys still alive past the state quarterfinals, so it ceased to be a problem.
From there we were active pretty much the next 7 weeks, intensively. Then we took a break in August and resumed for 6-7 weekends in the fall. We could've played more but we wanted to leave time for players to make college visits, attend a showcase here and there, or go to a college's camp.
The main thing is, you need to have a roster size that strikes the right balance between having enough arms and guys to fill out a lineup card if a few are hurt or absent, on the one hand, and making sure everyone gets the playing time/exposure they need, on the other.
One thing you can do is encourage your best pitchers to focus on being just pitchers. Then you have some athletes available to fill in if a position guy is unavailable for any reason.
There was another thread that had a discussion about the difficulty of being a 2-way guy in college. It's a fact that very few pull it off. While it's fine for a kid to do both for his HS team, for college recruiting trying to serve both masters can create confusion as to what it is he's trying to do and at what positions he's being projected. If you can education your best arms that they are best served on the travel circuit by focusing on pitching, and get them to accept that role for you, then you have built in bench players in the bargain who'll always appreciate the occasional at bat.