As someone who spent my entire twenties being rejected for jobs because I was "overqualified", I sympathise (I temped for a long time and then ended up in the NHS for a long time, which was a very good fit - they're the biggest employer in the UK, and new jobs are advertised all the time on the NHS jobs website. Even where I live, which is a very poor West Country town, in the last week alone there've been about a dozen part-time reception and admin/secretarial posts advertised. For health reasons I'm not in a position to work outside the home at the moment but I'm hoping to work back up to it and despite all the issues the NHS would be high up my list to investigate, especially having worked for them previously).
I used to get thoroughly exasperated by the overqualified feedback; I still needed to eat, no matter what my educational background. So many of the managers interviewing me had no clue what it was like to graduate into a recession (the last one, not this one), mostly they'd gone into their jobs at 16 and been ever so secure, so they thought. I do wonder how many of them are now facing redundancy and the reality of the high street.
So much depends on the local market - if you live somewhere like the north east where unemployment is very, very high, it's going to be much, much harder to get a job, any job, than if you're in an affluent market town somewhere where many people already have "professional" jobs or inherited wealth and don't need part-time work, so the competition is far less. So it could have been that you just weren't a good fit for that particular branch at that particular time, but don't give up.
Where I live the main weekend posts I've seen advertised have included estate agents' receptionists, care work, and, seasonally, hotel work (bar work and restaurant and cinema work and jobs in places like Costa etc all year round. Some of our coffee shops don't advertise in the paper or online though, they just put cards up in their windows to say Help Wanted.). Very occasionally things like weekend library assistants will get advertised, though they really are like gold dust these days.
Good luck, and don't give up - if you want a bit of encouragement go and look up the story of Johnny Cash, who sold electronic devices door-to-door for years before he became a professional singer. His take on salesmanship is really worth looking at.