Actually Kuryakin became a "full-fledged partner" of Solo's rather early on in the first season, because viewer fan mail demanded more of him -- particularly female viewers smitten with David McCallum. Vaughn and McCallum made a great team, and they worked well together.
Dr. Kildare was in color by the last season (1965 to 66). Although I've never seen the show, I know that because I once had some old TV GUIDE issues from that period, and at that time all shows in color were designated with the word COLOR in a rectangle next to the name of the show in the listings. It was also during that time when the series was seen two nights a week as a 1/2 hour program instead of one night as a one hour show as it had previously. That's why there were 58 episodes in the 5th season. Judging by the TV Tome website, one other thing that's notable about the final year is that it looks like this was one of the earliest series to have multi-part story arcs.
From what I read on NBC everything on their prime-time schedule in the 1965-1966 season was shot in color except for the first season of
I Dream of Jeannie and a failed WWII action drame called
Convoy.
Norman Felton from the U.N.C.L.E. series was Executive Producer of Dr. Kildare as well!

Other than that, a black & white series switching to color film was always a good move both professionally (in terms of ad revenue) and artistically. It certainly benefitted
The Fugitive to an extent, as did increasing the appearances by Bill Raisch as Fred Johnson -- the one-armed man.
Two WWII series seen in B & W for a few years were in color for their final seasons:
Combat! and Quinn Martin's
12 O'Clock High. They were both much more realistic in color, although ratings aside it was tougher for the producers to find much color film from WWII. It exists (thanks to such notable filmmakers as George Stevens and John Ford), but stock footage doesn't work as well in B & W or color. Re-enacting aerial battles is the best way to go, but in 1960s television there just wasn't a weekly budget for that sort of thing.