Chris Sunami is right-on: soul is the secular continuation of the kind of gospel music you describe, and if you're interested in not being limited to religious music, it's the place to start.
It's true that soul is a broad genre, but as opposed to say, electronic music, it's not something whose fans have delineated into highly specific subgenres. Rather, the styles of soul are often associated with specific labels (Stax, Motown, Atlantic…) or places (Memphis, New Orleans, Detroit, Chicago, Philadelphia…) Also, some notable soul musicians started their own record labels, or otherwise developed a community of performers whose music is of the same basic style. (For example, James Brown, Curtis Mayfield, Al Green…) It's also the case that most of the major soul musicians recorded diversely: uptempo and slow music. If you really don't care for the slow stuff, I think you just have to sample around.
This may be a case where, rather than searching for genres, you should read up on the history of the music. Articles and books about soul will relate music you already know you like to music you haven't heard yet. From there, you could start a Pandora or Rdio station that would play you lots of related music. Peter Guralnick's Sweet Soul Music is a great general starting place, especially in line with what you've already indicated you like. It doesn't cover New Orleans or any Northern soul, but what it does it does very well.
Another angle: this was music from a "singles" era, so there are a lot of compilations, some of which will lead you to specific artists you love. I learned a lot about soul from Rhino's Beg, Scream, and Shout— it is a great comprehensive introduction to this music. It's out of print, I see it listed used on Amazon for about $70, which is a fair price for a six disc set. I see that some but not all of the volumes are on the iTunes store and Rdio, so you can sample that way.
Here are some more things you can stream to sample:
Anything like this is hopelessly incomplete, so while it would be fun to go on, I have to stop. Hopefully it's a starting point. Also, referring back to the kind of gospel you described at the beginning, a record which I love but which I can't find streaming is None But the Righteous: Chess Gospel Greats
Addendum
Finally, while an introductory answer has to focus on the best-known artists, there's a lot of other great stuff to hear, and you don't have to dig as hard for it as people did 10 or 15 years ago. There are great labels reissuing overlooked gems from the 60's and 70's (often with a single-city focus). Check out Numero Group's Eccentric Soul and Good God! series, as well as Now Again and Light in the Attic.
And there are current artists making new music that ought to be heard. Being from Chicago, I'm especially fond of JC Brooks and the Uptown Sound and of course, there are great ladies like Sharon Jones and Bettye Lavette. So don't stop with the dusties...
