Dan Eubanks is amazing, isn't he? Great guy too.
IMO, a problem w/ tunes like St Anne's Reel and Whiskey, is the darned fiddlers tend to play them so darned fast...! They are a couple that I regularly feel quite solid on at home, but then when the fiddlers "put a little air under it"...
One that works out well for me is Old Joe Clark. I also like Harvest Home. Has a tricky descending run at the end of both the A and B, but it is not too hard to figure out - and you can always drop a note here or there. If this is your sort of thing, Alfred has
a series of books that have dots in bass clef for a bunch of trad tunes.
Others that seem to be played at a moderate pace include Flop Eared Mule and Nail that Catfish. A nice slow melodic one is Ashokan Farewell.
Are you looking to play them down in 1st position, or up at the heel? A lot of the tunes lay out really nicely up in 4th - but IME those also tend to be some of the tunes that tend to be played the fastest. And - of course - they require that you be comfortable up the neck (not something many BG bassists are comfortable with.)
The folk I play with enjoy it when I take a break. In fact, I kinda had to learn a few defensively, as they would tend to yell, "Take, it, Ed!" - and then drop out while I threw up all over myself.
Finally, work on all purpose patterns in your basic keys, to make up a 1/4/5 breaks that you can throw into most standards. Find a couple of really straightforward recordings that you can tune to (avoid old Flatt and Scruggs!

) and just practice soloing over the whole thing.
(BTW - "head" is not a term I've ever used/heard in BG. I always assumed it was a jazz term, which I was never sure exactly whether it meant the intro, or what. How are you using it?)