All of the left handed relievers who were non-tendered should get a look. With the three batter minimum, they need to be able to have some success against right handed hitters, though.
Given the Twins roster construction, they already have three utility types (Gonzalez, Adrianza & Astudillo) and a regular for every position except first base (or third if Sanó is moved). A right handed hitting outfielder who can play center field well (as well or better than Kepler) would be a good add.
There were also a number of catchers non-tendered, although none of them hit left handed. If the Twins are interested in Garver getting reps at first, they should be interested in adding major league catching.
Like the way you are thinking here! Agreed on checking out various LHRP options. My dream was Pomeranz on a 2yr $4-5M deal but that got completely blown up. Now, it's about who is the best candidate Johnson and the FO like. I kind of hate to say it, but I'm starting to wonder about Perez. I think his days as a SP option are gone, unless someone decides they can still "fix" him the way it looked like the Twins did the first couple of months last year. If he would accept a bullpen move, I find him interesting.
With all due respect to Cave, who I like despite some rough edges and a play in the field here and there that left me scratching my head, I'd be very interested in a lower cost 4th OF who can play a legitimate CF. I don't care if he bats LF/RF necessarily, just a guy who can play good defense and be solid offensively. I just don't know who to target at this point.
Catcher is weird and interesting. There are a couple guys I like...really liked Vogt...but still prefer to go status quo with Castro. Interesting I haven't heard a single rumor about him. Leads me to believe a re-sign is a strong possibility.