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For your situation, you'll want to check to see that your database is still referencing the same URL structure as well (look for things like switching from www to non-www as default or HTTP vs. HTTPS.)
As Sean says, focus on URL structures. If old URLs are working just the same as the ones on the new server, things should be fine.
Do you see any crawl errors in Google Search Console? That will give you some of the best insights.