How about:
Tell me something good by Rufus
Papa’s in bed with his britches on by Cab Calloway
Cecila by SImon and Garfunkle
Tender Trap by Frank Sinatra
Someone to Watch Over Me - I forget the original artist… i think maybe Cole Porter?
I’ve mentioned this band before, but maybe something like “She Satisfies” or “Karen” from The Shoes album Tongue Twister.
Their earlier albums, Black Vinyl Shoes and Present Tense have plenty of coverable classics, yet they’ve managed to remain pretty obscure for all these years. Also, what makes them great to cover is that their arrangements are sparse, their melodies are strong, and their songwriting is straight-ahead pop-rock in the style of, say, The Beatles or The Raspberries.
There’s also The Rezillos. A song like “(My Baby Does) Good Sculptures” is another straight-ahead rock-and-roll/punkish number which would work well for interpretation, and it’s silly, whimsical, great for a party atmosphere.
I can think of plenty more, but those are my two for now. I haven’t heard the Ataris cover, but I think it’s generally pointless (unless you’re a bar/party band) to redo a song as closely as possible to the original.
What TeaRoses says about the Boys of Summer is right. It’s a very close copy, but there is an underlining edginess to it. Basically, it highlights the great melody and lyrics.
I’ve never liked the original, but the Ataris have opened me up to the fact that it’s a very decent song.
I’ve always liked the selection of covers by Goldfinger.
Not sure if your band is the right type of music to make it work, but every day I dream of hearing a death metal version of “The Devil Went Down to Georgia”
Almost any classic blues song - although that’s obviously not the kind of band you are, they’re songs that cross genres and (in my opinion, anyway) stand up to all kinds of unusual interpretations. I’m talking Muddy Waters, Willie Dixon, Elmore James, etc. You cite the Stones as an influence, and they sure did their share of this, as did most bands from that era. And if Marilyn Manson can do I Put a Spell On You… That’s not necessarily obscure material, but I don’t see a lot of newer bands doing it. If you like Nirvana’s version of Leadbelly’s Where Did You Sleep Last Night? - which I think is absolutely gut-wrenching - you might give this a shot.
There are plenty of Dylan songs that might be worth a try, too… everybody covers his stuff, but there’s a lot of it, so who knows. And your singer is unlikely to be afraid of not sounding like Bob.
I’d like to hear a techno cover of Cream’s “Tales of Brave Ulysses,” a gangsta rap cover of the Beatles’ “I Am the Walrus,” and any cover of “Does Your Chewing Gum Lose Its Flavour on the Bedpost Overnight?”.
I once heard a really interesting cover of Sunshine of Your Love sung in very slow fashion by a woman - a search at the All Music Guide leaves me suspecting it was Ella Fitzgerald… that’s pretty mind-bending…