I was in the same predicament as you about two years ago. I feel ya
Last year of college and I started to look for part-time. If your town is anything like mine, it will take a while. Mine took a year, so I had to take a gap year 🧍
For some reasons, dumb a** local governments in small towns keep building lots of apartments without investing in the town itself. Stores are closing left-right-and-center here; the building prices are stupid; big corporations keep deciding, yeah, i need two stores in this town so let's take up space; the stupid it now costs to park 24/7 that discouraged people travelling here. And obviously this means there is not enough jobs to support people, but they're so out of touch and greedy that nothing will be done about it.
Look in the nearby towns. Sainsburys, Mcdonalds (do not recommend it, worse 3 month job of my life), Starbucks, Costa, Waitrose, Home Bargains, Holland and Barrett are a few names off the top of my head that I know don't ask for experience. Waitrose, Costa and Home Bargains you need to directly look on their job website to see vacancies. Many other businesses are like this and as the other post said some don't advertise online. Mainly independent stores, but I've walked past a Waterstones in another town and saw the vacancy on their window saying to ask for email inside, checked later and there was nothing about it online so some big retailers do it too. Some do it online but don't use indeed, like Robert Dyas.
Redo your CV. A lot of my friends had really bad CV's, honestly. Plain and boring and ordinary and not very informative. Just stated their education and work experience and a few short bullet points of hobby and skills. I helped my friends build a new one by making it and structuring it on word and then asking for the relevant information, and ended up having to make one for my older sisters friend, too. So, don't be afraid to ask people for help :]
I had been out of college for a few months and had been ghosted 95% of the time by jobs (seriously, why do they expect us to be professional when they aren't themselves?) before I applied to Home Bargains Evening Team. Got an interview and was hired on the spot, so they must have been desperate and sometimes it'll be like that. But, I am being paid £13.55 an hour with an extra £1 each hour due to being in the London region, so I am not complaining. Got it by a stroke of luck of being there at the right time, and the job is nice with a smaller and close knit team than the day shift.
But, yeah, it took a whole year.