This is the first I’ve ever heard about keeping e-kon in the fridge… cold temperatures usually decrease the stickiness, and I would worry you would lose too much stickiness in the fridge. Being non-sticky is a bigger issue than being too sticky, because while it’s easier to get off the roll, it won’t do it’s job.
I do try to avoid extreme temperatures. As you know, heat increases the stickiness (pro tip: if you really need it to stick when applied, pop it in the microwave for a few seconds), so it can be hard to get off the roll in summer heat. Likewise, it can be useless if trying to apply it when it’s too cold, like if it’s been stored in a frigid barn in the wintertime.
I always keep at least one roll on hand. It’s rare I have more than one roll, though, because it’s expensive and I don’t use much of it at once. I generally just use it at the tops and sometimes bottoms of bandages to keep them in place and keep debris out, which means I’m only cutting a few strips off at a time. I love it for applying around the pastern/coronet to keep hoof bandages in place.
Re: vet wrap-- I always keep vet wrap on hand… I buy it any time it’s on sale. I get antsy if my stock drops as low as a half dozen rolls. But vet wrap doesn’t have adhesive, so it isn’t affected by temperature. E-kon is a luxury and you technically could wait to buy it in an emergency (if you have a local source), but vet wrap is not IMO.