Acclimating the strings/bass to the room is the key. Even if you can't get the bass out of the bag until 15-20 minutes before downbeat, if you play it and keep tuning as you and the bass warm up, everything will be much more stable on the bandstand. I keep a tuner in line as well and sometimes the first half of the first set is a lot of tuning. Mostly though it's just the first song of each set.
On a separate note, I had my Dlugolecki E string rewound by Larson recently. It sounded really bright at first but it has settled in nicely. The rewound string is ever so slightly more direct sounding because I think Dlugolecki actually wraps his bass strings twice with wire and Larson seems to have used a larger diameter wire and only used one wrap. The final string gauge is the same.
-J
On a separate note, I had my Dlugolecki E string rewound by Larson recently. It sounded really bright at first but it has settled in nicely. The rewound string is ever so slightly more direct sounding because I think Dlugolecki actually wraps his bass strings twice with wire and Larson seems to have used a larger diameter wire and only used one wrap. The final string gauge is the same.
-J
then I'll play a few minutes, tune again, and repeat if necessary. By this time the bass is usually stable. Guts definitely respond to climate change, but once you get them stretched and warmed up in any given time period, they don't really move that much. After a while, you start to think like a gut string and you can outsmart the sucker 
Listen to it sustaining!
