How should I blend floor stain applied to different areas at different times?
I've had pretty good luck feathering in paint, but stain is hard to do. I've never had a perfect job. I've only done homemade furniture and often choose the rustic look so it's not critical to get it perfect. If the seam is at the threshold (hopefully it is) you can stain the rest of the floor as usual getting full coverage without overlapping as most stains will darken when restained. At the seam, stain over an overlapping swath with a heavily diluted solution.

Test First
If possible use a scrap of the wood to test the procedure. If you don't have a scrap of wood buy some of the same wood if you can. 

Apply the stain to the scrap let dry. And then try your method. Do multiple tests. And check it in various lighting and angles. 

This countertop was blended in. It's hard to see from the image but it is noticeable 

It's a pine counter top I made from stuff lying around.