I'm surprised no one has responded to this yet.
If you like The Welcome Wagon, I'm going to go ahead and assume you're familiar with Sufjan Stevens. He's a Christian and particularly his early albums are influenced by that (Michigan, Seven Swans).
Wovenhand (also sometimes typed Woven Hand) -- dark, textured folk
--if you like that, the lead David Eugene Edwards is also behind 16 Horsepower, who comes highly recommended as well, although 16HP has some more adventuresome lyrics.
twothirtyeight -- light, dancing guitars. Shins-esque vocals. Came out pre-Shins, so this is no knockoff.
Starflyer 59 -- prolific shoegaze.
Sal Paradise -- mid-90s alternative, seems to tell a story as much through the guitars as through the lyrics.
Roadside Monument -- Odd little group where albums balance on the precipice of a hardcore leaning, but there is so much great indie-rock-styled guitar-work that it's hard keep them off the list. Their album I Am The Day of Current Taste is probably the best starting point, as it is their last album and, as such, the one which takes the most mellow approach.
Over the Rhine -- folk/alternative.
Michael Knott (and LSU/Lifesavers Underground/etc) -- prolific 90s/early-00s singer/songwriter. Some of his early albums (Grace Shaker) were some of the darkest alternative Christian music at the time. Lyrically stunning stuff--he tells excellent stories through song.
Luxury -- Each album feels different. "Luxury" features dark, swirling guitars. That is my pick for their best album. But "The Latest and Greatest" is also good.
Lost Dogs -- if you don't mind some alt-country, The Lost Dogs write some superb songs.
Kerith Ravine -- I guess this would be considered emo, but it came out before emo exploded into the mainstream, so it lacks all the generic absurdities that the genre has become since.
House of Heroes -- I can't speak for their most recent stuff, but their earlier albums are a treat. Almost feels too mainstream in sound to count for a list like this, but their songs are just so good it doesn't matter. Especially the self-titled release (later spruced up and re-titled Say No More. If you can find the self-titled version, I recommend that. But Say No More isn't so much different to be worth spending time hunting for the original.)
Havalina Rail Co. -- Havalina parades around a mixture of styles, starting off more swing in the early years, before covering folkier territories. Their album "Russian Lullabies" is, for my money, a Christian masterpiece.
Eisley -- Chick singers, fairy-feeling music, can't go wrong.
Dennison Witmer -- singer/songwriter, though the only album I've found worthwhile is "Are You A Dreamer?", which I think is available for download free through noisetrade.com.
Dear Ephesus -- The production feels really dated 10-odd years later, but Dear Ephesus is in the line of complexly composed, interesting story-telling vein of alternative. Slips a bit into emo, but again, pre-Mainstream stuff.
The Danielson Famile (also Danielson, Brother Danielson) -- Either you get it or you don't get it. Think of a mix between The Welcome Wagon/Sufjan Stevens/Neutral Milk Hotel and a Helium baloon. One of the most creatively stunning groups to trapse around the Christian scene--and one of the most misunderstood.
Cush -- The best of the Michael Knott-fronted projects, Cush was intended as a supergroup of Christian 90s alternative "stars", except they kept their identities secret. Unlike most supergroup type bands, Cush harnessed their varied talents into one of the great Christian alternative albums. (If you look for it it's the self-titled one with a skier on the cover)
There are others, but this is a good start, anyway.
