Here is a comparison photo of a Smith&Wesson Model 1, 2nd issue with a Rollin White-type .22. The RW is marked "Made for Smith & Wesson by Lowell Arms Co." Note that it has a solid frame with a side loading gate and a plunger ejector. Some did not have the loading gate or ejector rod.
Smith and Rollin White.jpg
In my gun show experience these are pretty rare. Those with loading gates and ejectors are worth the most. Asking prices I have seen are usually above $500, sometimes considerably above.