The old site
Paste in a crawl export or a simple list of URLs. You can use a Screaming Frog export as it is, but keep the Address, Title 1, H1-1 and Meta Description 1 columns for more accurate matches. This lets the tool compare what each page is about, rather than relying on the URL alone.
The new site
Add a crawl export or URL list for the new site in the same way. Staging URLs are fine, and you can replace the staging domain with the live domain in the finished redirect map.
A redirect map is a decision about every URL you have, not a bulk find and replace. The pages on the decisions list are the ones worth your time: they are usually the pages that no longer exist, and the honest answer for them is either the nearest useful page or a deliberate 404.