I would drop her a note with a brief resume of what/where you have taught before and saying that you would be available at short notice if she needs you. This avoids putting her on the spot, and gives her the information she needs on your background etc., and she may well then call you in for a chat. If she doesn't need you, she may be able to suggest other local schools you could try.
I did, and it was fine, but then I have 3 children in the school, have been on the PTFA since the eldest started 6 years ago, been a parent governor for 4 years and have helped out in school whenever they have needed! She was pleased I approached her, as she had considered asking me, but didn't want to put me on the spot!
If you don't know her that well, then an up to date cv would probably be a good thing to have available.
She's emailed me in the past and asked me to look over a policy related to a health condition, but she's not friendly if that makes sense? Probably best to do a cv and letter and hand it to the secretary.