I’ve actually done something like this, albeit not due to impatience. The reasons are too long a story, so nevermind, but I’ve found that if I make coffee and immediately decant and cover it, then leave it on the counter, when I microwave it the next day (no refrigeration), add milk and Splenda, it tastes like I just brewed it, at least to me.
It’s the stuff that has been left on the warmer that tastes like crap, even if it hasn’t been that long-- albeit, the longer, the worse it tastes. Also, stuff that has been left uncovered is not good. Plus, I have taken coffee right off the warmer, not decanted it, and just left it on the counter-- tastes awful, I assume from residue left in the pot, but even stuff that has been decanted and left uncovered is not great. Refrigerated coffee, covered or uncovered, I can tell the difference, but covered and refrigerated is drinkable, it just tastes different.
If it’s had anything put into it-- milk in particular, but Splenda or Stevia as well-- it tastes “wrong” the next day, no matter what I do.
It’s worth noting, though, that I don’t buy the most expensive coffees available, and maybe the expensive ones are more delicate; also, I go for stuff on sale a lot, so that could mean it’s old. I have a Keurig, but I use a reusable filter, and put ground coffee into it, so it is quick-brewed. I also use flavored coffees a lot, and (sugar free) syrups, which could mask any staleness picked up from being left out.