That was the case in my recent project where a client wants to use his current hosting provider for email but generally, it was advisable to change hosting due to performance reasons.
The solution is easy if you have done it at least once :) It's enough to change domain A record to point the target hosting provider sever IP.
Example: Let's say I own grzegorowski.com and my hosting provider is myprovider.test with server IP 10.10.0.1, so my DNS record looks similar to:
Record Type: A Record Name: grzegorowski.com Target: 10.10.0.1
If I want to change only hosting provider to otherprovider.test with server IP 10.10.5.5 without changing email configuration, I will have to modify my DNS A record to:
Record Type: A Record Name: grzegorowski.com Target: 10.10.5.5
Of course, in order to serve a website from otherprovider.test server I will have to prepare SOA zone in his admin panel for the delegated grzegorowski.com domain (similarly to the regular full DNS delegation).
The SOA zone is generaly created by adding your target domain to the server admin panel.