Is there a way to mitigate any issues? What should I consider when migrating servers?
So long as both servers have a valid certificate for the domain (even the exact same one on both), and so long as everyone is still accessing the site via the domain name (i.e. going to example.com as opposed to 126.96.36.199), the SSL certificate is not affected, and neither is the user experience.
In fact, hosting your site on two different servers, each with the same certificate, while awaiting the DNS change to move from one to the other is probably the most transparent and user-friendly way to migrate servers -- 99.9% of your users simply will not know the difference (provided there's no errors or anything on the new server, of course).