I will typically do as Skye described above, and then build a report for myself that shows every instance of the old field that's been filled out with the new field side by side. If I have minimal data/records entered, I'll just get a cup of coffee and copy/paste for a few minutes.
If I have extensive data that copy/paste isn't reasonably feasible, then I use a Fusion loop to get all of the records with data in that field and then copy it into the new field. I do NOT use Fusion to remove data from the old field though. I figure if the copy portion goes sideways, I've got an annoying mess to clean-up. If the delete portion does something unexpected, I've got a lot of irritated people and not enough coffee.
Downloading the original dataset before starting any of this can also sometimes help mitigate hiccups.