Furniture, Wood and Cabinetry Finishing - Stain seeping up through wood

I am begging to wonder why I have taken on this project! I am refinishing to chairs that were made in Checkoslavacia 100 or more years ago is my guess. The wood is maple. The original stain was a dark red-mahagony. I have stripped off all the varnish and sanded the chairs. The stain was very difficult to sand off. I removed as much as I could and then bleached them. I used good old clorox. The chairs would look like the stain was all gone and then I would neutralize them with a water vinegar solution and then rinse them. Later afteer they had dried I would take a look at them and they resembled the inside of a peach with this speckeled look to them and it almost seemed like the stain had run. Should I continue to bleach them? Should I leave the bleach on overnight. Do I have to neutralize the clorox bleach. Help any other hints as to what I can do to get rid of these speckles? If I sand this and it seems to go away, will it show up when I put on the new stain? I have no idea what kind of stain was originally used but it sure seems to have soaked in well. One chair has only a few speckled places left. HELP!!!

You were on the right track - but Clorox isn't what you need to really clean these up.

Get a two-part bleach - look for that description. It's two containers, hydrogen peroxide and sodium hydroxide. Mixed together they form a powerful oxidizing agent. It's the strongest available (for wood) that I know off.

Neutralizing the bleach is NOT necessary. After bleaching, simply wipe down with a wet rag, then wipe dry.