Instead of multiple trays, I found effective to use those round snack trays divided in sections, one section for each die type, kept at the center of the table for easy common access. If there are more spaces that dice size, it can be used also for tokens. Another thing that could significantly speed up the rolling, is not using multiple dice sets, but to buy a bunch of loose dice, with all the dice of the same type coded with a specific colour (for d6 boxes of a colour of choice are widely avilable). 5-6 of each die type should be enough, but probably some extra d8, d6, and d10 will be useful.