Last week of June/first week of July is a safe bet in a "typical" year, though stream crossings will still be a bit big. Big difference between passes like Spotted Bear, which is lowish, gentle and faces south, and White River, which is quite steep and E/W. White is probably the last trailed pass on the Continental Divide to melt out.
Late summer is a better bet across the board, unless you're packrafting. The weather is more reliable, streams are lower, fishing is good, bugs are gone, and so forth.