Oil can work nicely, on high quality leather. However, cheap, chemically tanned (even "drum tanned") leather will simply soak up the oil, which will weaken the fibers, making it stretch and break more easily as time goes by. It can also soak up a LOT of oil, taking days and days, if not weeks, for all of the "excess" oil to come out of the leather.
The best way to have "soft" leather that will last forever is to spend the money on QUALITY leather goods made in North America, Europe, or Australia, and some Argentine leather goods. Stay AWAY from any leather products made in India, Pakistan, or any other part of Asia. Even those made with "English bridle leather" are still only "so-so" in quality.
Example: I just bought a pair of "Silverleaf" reins and an "Ansley" dropped noseband from Toklat. Silverleaf is supposed to be Toklat's new "premium" leather goods line. Toklat - Horse Tack - Saddle Pads - Horse Riding Apparel - Tack & Accessories - Bridle Parts
I was not impressed... Both had tags "made in India" and the leather was not near the quality I was expecting for the price.
I also just bought a pair of Bobby's reins, used from an HGS forum member. THOSE were very nice, high quality reins (barely used, soft and supple!). I've just ordered a Bobby's bridle from a member here, and I'm expecting it will be just as nice.
Bobby's tack isn't as high priced as some, about the same price as Ansley and the new "Silverleaf" by Toklat, but the quality is so much nicer.
I use warmed Hydrophane Leather Conditioner (comes in a can, oily) or Passier Lederbalsam to soften and preserve leather. I prefer the Passier conditioner as it won't "over condition" like oils can. You just whipe off the excess and on you go. Plus a little goes a long way, making it a very economical leather conditioner.