If you're looking simply to spiff the revolver back up a bit, save the time and money and buy some gray Norton scouring pads -- they'll most closely approximate the factory satin finish; run a search, there are many thread here on how to use them to restore stainless.
Mothers Mag, Flitz and such will get you a mirrored high polish; if you like the shiny chrome look, there you go. The factory high polish is the same: mirrored, chrome looking.; it isn't the brushed, semi-reflective finish the revolver left the factory with originally.
Most refinishes don't help value for collectibles expect under certain conditions and usually for much older guns; they sometimes can for non-collectibles, if the buyer appreciates the refinish and wants it, but don't expect to recoup the full cost of a refinish.
The 3" 66 is a sought after piece, but as a carried and shot gun minus factory box, tools and docs, isn't going to merit much consideration by collectors. With that in mind, you can do just about whatever you want to the finish and the piece will retain value to shooters.
What's your intended use? If it's to be carried and shot much, whatever you do with the finish (short of applying a coating) will wear again, in which case it might not be worth the time and cost to send it to the factory, or do anything at all, for that matter. But if you're going to sit on it and want it pretty -- which is fine and many of us do -- you can try the home-based approach with polish and pads, or go the factory route for more cost.
Although not listed on their site, the factory can do a brushed, original finish on it, not just the bead blast or high polish.