Get the counterpart from A Shot in the Dark as well.
Yes, Johnny Rivers is the version I got from iTunes.
That’s the one they used on the animated The Inspector, right? That’s a good one. I’ll d/l that one for sure.
The 3 I hum while sneaking around are:
James Bond theme variations,
Mission: Impossible,
Theme from S.W.A.T.
Don’t forget:
"The Last of the Secret Agents (And He’s My Man) by Nancy Sinatra
“Casino Royale” by Herb Alpert and The Tijuana Brass
“Theme from The Man From U.N.C.L.E.”
The Inspector Gadget theme?
“Spy” by They Might Be Giants?
For the smokey back alley sound (maybe more PI than spy) – Harlem Nocturne, commonly remembered as the theme from Mike Hammer.
The theme that immediately stood out to me as being missing from your list is Man in a Suitcase.
You may also like The Champions, The Persuaders, and The Protectors (ah, Tony Christie).
More PI than spy, really, but Randall and Hopkirk [Deceased] is nicely atmospheric – you might even get a kick out of the 2000 remake.
And how can we forget the most iconic TV secret agent of them all? 
Probably the pioneer in the genre: Raymond Scott’s Secret Agent, a brassy cat-on-the-prowl item from the early '40s.
Sade Smooth Operator
Dionne Warwick sang a great Bond song,Kiss Kiss Bang Bang.They used the instumental of it in Thunderball. The Man from uncle was great also.Every year,Jerry Goldsmith would jazz it up a little more.Not exactly a spy theme,but Sledge Hammer was a great theme for a crummy show
You might want to add these James Bond themes:
Die Another Die - Madonna
The World is Not Enough - Garbage (already mentioned)
A View to A Kill - Duran Duran
For Your Eyes Only - Sheena Easton
Live and Let Die - Paul McCartney
Nobody Does It Better - Carly Simon
Goldfinger - Shirley Bassey
An obscure show of the kind you want was Checkmate, with, of all people, Sebastian Cabot. I can’t find the theme song, though. And at that link, would you believe Sebastian Cabot reads Bob Dylan? Shatner, move over.
I’ve got that one in a ska compilation. 
Some of those are good – and they were used in the Bond films – but not really what I’m looking for. Thanks, though! 
This may be a little off-tangent, but I just have to recommend the CD Sex Mob Does Bond. Probably my favourite jazz find of the last couple of years.
The cover caught my eye in the jazz section of a small record store and I immediately bought it based on two factors 1) The title and 2) I’m a big fan of John Barry’s music. I was blown away when I threw it into my car’s CD player on the way home. It’s basically a reworking of several Bond themes in stripped down arrangements that let the players burn through some furious killer improvs. Many tunes have a soulful backing chorus and snare beats that really emphasize the cheesy sexiness of the original tunes.
I can’t recommend this CD highly enough, esp. for jazz or John Barry fans.
Agent double-o soul?