First off, Pinnacle Entertainment Group changed the way that Shaken works back in May 2015. Now it only requires a simple success (TN 4) to get out of Shaken, and there is no longer a "recovered from Shaken, but can't act this round" condition. In fact getting a raise no longer does anything. That will make things easier. (For more info on how this affects the game, see How does the change to the Shaken mechanic affect other parts of the system?)
Even then, it's not a gaurantee to get out of Shaken. So there are a couple of things you can do to help your party
Use a cooperative roll
Come on, snap out of it!
This method is always available, so I'm saying it first. Just go on hold, then when the other player's turn comes up, you can act by making a cooperative Spirit roll. Make a Spirit roll yourself and for every success and raise, you give the other person a +1 bonus to their Spirit roll (max +4).
Take the Command Edge
I'm not giving up on you. Don't you dare give up on me!
This Edge provides a +1 bonus to recover from Shaken for allies within 5" of you (that is, 5 battle mat inches, or about 30 feet/10 meters).
Use Boost/Lower Trait
Full house: Queens over Sixes. I've dealt with you, Devil, now give her spirit some supernatural aid!
If you have magic available to you, you can use the Boost/Lower Trait power to increase the target's Spirit die type. With a success, it goes up one die type, with a raise it goes up two.
Heal the character
Rub this salve blessed by the spirits over your wounds. You'll feel a lot better.
You mentioned that one of the main reasons why you would fail a Spirit roll would be if the character is wounded. So obviously, healing them would help them. Note that only magical healing (or since this is Deadlands, a potion from Smith & Robards), is fast enough to work in combat. Non-magical healing through the Healing skill takes 10 minutes per use.
Give them a Benny
When you're at your lowest point, remember that I'm fighting right alongside you.
If a character takes the Common Bond Edge, they can hand over a Benny to the affected player, which they can immediately spend.
Let them have a certain Coup that gives a Benny
Soaring Eagle said that in ages past, the braves ate the hearts of the defeated. You've had so much weirdness happen to you that maybe for you, it would be true.
Since this is Deadlands, a character might be Harrowed and can take coup from defeated monsters. While kind of a roundabout way, there is one monster that gives you an extra Benny if you take its coup:
By letting the Harrowed character get an extra Benny, they can get out of Shaken on their own one more time.