The cleaner you can get the top the better the final finish will be.
Hang in there!
Linda C

You talked about removing the finish(es) and then referred to the stain remaining. Strippers seldon remove stain, they are designed to remove finishes that are on the top of the wood, not in the fibers of the wood like stains.

If there is still stain present, any further staining will be affected by the existing stains. If the finish is gone---a third application of stripper should make that as sure as possible---and there is still color variations, you might try bleaching. I have never tried that, so can't comment. But there is literature describing how/what to do.

Thanks for the input. I've been using "Peel-away" stripper which is probably mild since it's sold in a plastic bucket, so maybe I need to do a few more. The stain was partially removed on the second application, since my wipe-rags were covered with the reddish-brown color. I'm too afraid to try bleaching, since the wood is absoulutely beautiful, and I don't want to goof it up terribly.(Boy, they don't make tables like they used to.)

Bleaching shouldn't affect the wood or the grain

Here is a link that might be useful: www.rcfurnture.net

To the OP. It sounds like you've removed the finish , that's what stripping does . It removes whatever the clear product is on top of the stain or natural wood.
The next step is to sand the wood, this will remove the old stain that has been absorbed into the wood.You must sand the entire piece , not just the sections you mentioned.
I would suggest at the very least starting with a 220 grit paper by hand. I do not know if you have a random orbital sander. If you do , start with 120 grit with the machine and then follow it up with 220 grit by hand.
This will provide you with the proper surface to either stain or finish natural.

Stocky: I don't have an type of mechanical sander. How do I know "how much" to sand (ie, when to stop)? Just like a novice cook may not know how to precisely time when to stop baking/roasting a piece of food, I'm still unsure about this stuff. Do I keep sanding until the wood is nearly "natural", like a fresh piece of lumber? Thanks

Thanks for your help. I'm pretty sure the table is maple, but with that reddish-brown hue still left in some parts of the wood, maybe it's cherry (but no knots, and a fine grain). The top of the table (without the leaf) is made of 6 planks of the wood about one inch thick. I'm so excited about restoring this table. My father, who passed away a few years ago, refinished all our furniture in our house, so I also try to think back about how he did things.

You got the finish off but you have to sand down to get rid off the stain. This is a job for an inexpensive machine. Get a Bosch random orbital sander (the best and a good value) that uses 5" diameter hook and loop sanding disks. Start with 80 or 100 grit which is aggressive and you will relatively quickly even out the color variations. Move thru finer grits 120, 150 then 220 grit. This approach literally saves you 90% the time of hand sanding and produces a better result for the novice. After the wood is even and new looking apply an oil stain and then finish with poly or varnish. Don't brush on the finish, thin it slightly and wipe it on. Best finishing results if you build up lots of thin wipe-on coats rather than use a brush. Who wants brush marks on their table? Take your time on the finishing, build up 10 or 12 thin coats, you will be proud. Test the stain color on the bottom of the table top (sanded area) before you commit!

"...sand down to get rid off the stain."

Since maple is normally very pale anyway, bleaching is liable to be a lot better than simply removing further wood.

If you plan on a darker color for the new finish the variation may be covered anyway.