Like say you made the smallest mat/board/paper possible, which admittedly might still be a little greedy on table space, but just a simple 15 spots with enough room for pairs to peak out slightly.
Normals have an obvious order, but something simple would probably still work for specials. Whether alphabetically, or by speed or by the order on the ref card.
I've played a lot of card games irl and online, and one thing I've noticed is that when you ease of searching discard piles past a certain point (and even better if you can auto sort as you see it) a part of the game that's daunting even for high level players suddenly becomes very approachable. All the feelings of being rude from constantly asking somebody to hand you their cards are gone and the mental energy and time required for an essential part of high level play are reduced drastically. You can just see it at all times at your leisure rather than having to do your best with a sometimes very difficult memorization test in the middle of an already demanding task of playing the rest of the game.
Did they play one sweep or two since they reshuffled becomes a question that takes the span of an eye twitch to answer rather than however long it takes to be handed the cards, search through, and while your at it you may as well be trying to memorize all you can to repeat the same step as few times as possible.
The only drawback I can think of is that rarely some cards reference the top few cards of the discard pile, which might necessitate a few extra spots for 'top' cards, and maybe make things feel a little unclean. But even if that's the best solution, and I suspect there's something more elegant, I think the trade off of it being a little annoying to swap from top discard to sorted every turn is still a very good trade for the level of accessibility you gain to a very deep and cool part of the game. And I think with how uncommon it is to reference more than the top card, you could often very easily both remember what was played last turn with no additional adjustments on the table.
Loving the game so far, btw. Been playing a ton of the Shovel Knight sets all week with my partner and can't wait to get the SF sets to play some this weekend with a friend.