I know this is in jest but they will gladly take a .500 June/July record and roll the dice at the deadline.
Montas and Contreras will be the hot trade candidates. I don't expect Cardinals to touch them (Cards being able to trade with Cubs will cost "extra" anyways. Likely out on Miley too). Eovaldi depending if Sawx are out of it.
If the Dodgers don't mind trading to a competitor in the NL, I could see David Price being the most likely candidate. He's FA next year and Cardinals are infatuated with him. Gibson could be an interesting middle tier guy from Philly. Lorenzen from the Angels. Michael Wacha return???