These "New Baker" trigger-plate hammer doubles based on A.C. McFarland's Patent No. 370,966 granted Oct 4, 1887, were introduced by Syracuse Forging & Gun Co. in 1887. The factory in Syracuse burned in 1888 and the company moved to Batavia, NY. The company name was changed to the Baker Gun & Forging Co. in April 1890. There is a very thorough article by William M. Furnish and Daryl D. Hallquist on Baker hammer doubles in The Double Gun Journal, Volume Nine, Issue 2, pages 63 to 77. Early on these guns had an underbolt and a dollshead rib extension. About March of 1892, they began bolting the gun with a wedge thru a slot in the rib extension. According to the graph in the DGJ article your gun was made about 1894. This model was made thru 1895, then for 1896, they had a diamond engraved on the trigger guard and were marked Model 1896. In 1897 BG&FCo. brought out there sidelock Model 1897 hammer double which was more of a hammer version of their hammerless doubles.
This info has been hugely useful for me and a friend who are trying to identify an old shotgun he has. We know for sure it is a Baker Gun & Forging Company shotgun, it is not hammerless, and in fact seems to be very close to the one the original poster was asking about. The serial number on this particular gun is 19801. Neither of us knows too much about shotguns, so some of the terminology is way over our heads here...we would really like to know, as close as possible, what year this gun was made.
The only other info I can provide is that the sideplates are very plain, lacking in any engraving other than "New Baker" which is very faint on both sides. The barrel appears to be a Damascus twist, but I am not 100% sure what that term refers to, so I am not totally convinced this is the case.
Is there any chance anyone here can help us narrow this down? The gun is rough shape, so value is not so much the concern; this is a family heirloom, so the info is more for historical knowledge than anything.