I got my current, part time, job a couple of years ago. It was the first I'd applied for after over 10 years as a SAHM. It's school based, support staff in an office.
I was running my own, very small, on line craft business and did some volunteering at the local Church. Both of those roles had helped me keep on top of Excel, Word and email. Both had meant I needed to be proficient in social media and dabble in marketing.
I made sure I matched both my past experience and my current skills very closely in the application. So I pulled out things I'd done in my pre-SAHM role and how I currently used that knowledge whilst a SAHM. For example, the role had a desirable rating for dealing with parents and children. I'd been a parent reader at primary school, had organised summer days of craft etc at the local church and in my previous role I'd done presentations on council run projects to youth groups, schools and communities in general. So I pulled that together as a 'I can do this because' statement.
The role needed marketing skills so I, again, pulled up how I market my small business, how I helped to market church run events and how I marketed community events in my previous (10 years plus ago) role. I also put things in the statement like ideas of how I could transfer those skills to this job - what I'd learned, how I'd proceed etc. The ideas may not actually work in the environment I'm now in but my manager said the fact that I'd thought about what I could actually bring into the job made the difference - it showed enthusiasm and pride in a job I didn't actually have yet.
So I'd say make sure your Microsoft Office skills are up to date, think about previous roles and what you learned in them but don't knock the things you have to do as a parent (stay at home or otherwise). The organisation, the negotiation, the teaching, the 'thinking outside the box' that we all do when we have a stroppy kid who doesn't want to get up and somewhere to be in 20 minutes. All that negotiation and tactics is invaluable when you have a worried patient who's getting in a state or someone who's been kept waiting who is kicking off (rightly or wrongly), if you want to get into the NHS.
And finally, yes the job market is tough. It was tough (but maybe not as much) a couple of years ago and school term time job are like gold dust. But as my DH said at the time, someone has to get that job and it may as well be you.
Good luck!