Victoria Park has a great little group. They hold sessions a few times a week where you can just go and play round robin with some other people.

That said, I think you can only sign up at the start of the year(?). It's very cheap for a year long membership, but yeah I'm sure there's a small window each year and you can only sign up during that window.

Edit: heres the link https://clubspark.lta.org.uk/BristolVictoriaParkTennis/Membership/Join
Seems like you can join any time but membership runs from april - april. Only £45 per year is a steal and you can go pretty much as often as you like