Interesting. Just appeared on ebay. I've done an internet search and have only come up with others also searching; no Baker Axe Co., etc. I did find a Baker Hardware in South Dakota and Baker & Hamilton, Agricultural Goods and Hardware, San Francisco.
I have no connection to the seller

http://www.ebay.com/itm/Rare-Baker-Axe-H...o-/131420903442





Hunter Arms Co. made water meters, electric motors and fans, and bicycles. Of course Winchester made tools and roller skates and about anything else that would sell. And J. Stevens Arms & Tool Co.
Any other turn-of-the-century gun makers also produce tools?