Hybrid email setup w/ managed server but self-hosted client

I'm planning to migrate my email to a different provider, but they don't give much storage, so I was wondering what people would recommend for this kind of setup: basically I'd like to use the new provider as something like a relay. I'd want them to only store an email or two at a time and have some kind of self hosted solution that just grabs the emails from the provider and stores them after deleting them off the provider so it's never storing my entire email history, and also keeps my sent emails somewhere so that I have a copy of it. Ideally I'd wanna be able to set this up with a mail client like NextCloud's.