I've done a lot of cycling over the years. When I went on a one week trip in southern Ontario I took big rear saddle bags and a small fanny pack. When I go out for a 30 minute ride in my neighborhood I take a bigger fanny pack. I've tried under seat bags but they never worked out for me.
It's
similar to this one.
It always loaded with a spare inner tube, tire levers, a screwdriver for gear cable adjustment, a couple of wrenches and alan keys for other adjustments and a couple of small rags. When I head out I put in my keys, small camera, phone and wallet. It also holds a couple of snacks and a PV if I feel like taking those.
I wear it behind me on the bike and for better access I can loosen it a bit and rotate it to the front if I feel like it.
Important things for me are the ability to loosen/tighten it easily, size and the pocket arrangement. My current pack has a main compartment just right for permanent residents and a couple more for the other stuff.
When I first started cycling I thought a bigger fanny pack like this one would be a problem. It's not for me and always having the essentials in there ready to go is great.
This works for me, but you'll have to experiment yourself to see what is good for you. The good thing is you can get cheap bags of all types to experiment with.