I'm looking for ways to access mysteries.
The list so far
-Oracle (duh)
-Speaker for the Past (shaman arch), time/ancestor only
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Actually, basic shamans (no archetype needed) get oracle-like revelations called 'spirit hexes' via their spirits. While most of the revelations that people dip oracle for are not available, there are some really fun ones in there. They also gain access to wandering spirits. This is a bonus spirit that they can change, with hexes from the wandering spirit coming in at level 6. This makes shamans very flexible.
Spirit guide oracles at level 3 can play with shaman wandering spirits, and that's when they also gain access to some abilities from other mysteries via the wandering spirit's hex. They can bond with a different wandering spirit daily. This is the only oracle archetype I know of that lets you borrow toys from other types of oracles... er, shamans.
You can also get access to some oracle goodies by variant multi-classing oracle but I would be wary of that option. They took out most of the revelations that people would dip oracle for, and everything worthwhile comes in at -6 to level. In my opinion, it's not worth it.
Things that won't work for non-oracles: ring of revelations and the soothsayer's raiment. You can't even UMD them. It is unclear whether even oracles can UMD them to gain access to revelations from other mysteries... that's something that has to be houseruled by a GM, and who would want to make hourly UMD checks, anyway?
TL/DR:
If you want access to an oracle's mystery, the best way to do it in Pathfinder is to dip oracle. There are many strong revelations that are available to first level oracles and then you can take more revelations with the "extra revelations" feat.
So... all that said... what oracle ability are you looking for? Is this question tied to a specific character that you are building?
I saw that Speaker of the past gave me access to oracle mysteries and revelations and changed charisma based things to wisdom. I was wondering if there was more that could do similar things. I don't have a set build in mind yet and was more looking for something to build around.