There's a lot of confusion about this. Just to clarify, people often say "OTC" when they mean "unlimited". OTC means "over the counter" of course, but what it really means is "not from the draw." You buy a tag "at the counter" (or online) - that's all. You can absolutely buy deer tags this way, and there are some available right now. I just bought one this morning.
With elk, the herds are so numerous and hunter success is so low that they have a set of GMUs (see Big Game Hunting Brochure here
http://cpw.state.co.us/aboutus/Pages/RegulationsBrochures.aspx, pages 34 and 36, anything in yellow) set aside as "unlimited" zones. It doesn't matter when you buy these, because they'll sell an unlimited number of them. You can buy them the week you drive in for the hunt. Or that morning. They'll be there.
The difference with deer is that there are no UNLIMITED units. All deer units have quotas. In normal circumstances you have three chances to get one. 1. The draw, but you've missed this for this year. 2. The leftover draw, also missed. (To get this you have to enter in the draw.) 3. The leftover lists. Those are what's "left over" after the draw, and you can buy them any time after they go on sale.
THAT'S TODAY. They're going fast and mostly sold out because they're in high demand but there are still some left! The leftover list is at
https://cpw.state.co.us/Documents/Leftover.pdf and updates every few minutes. They'll all be gone in the next day or two I'm sure but right now there are a few like DM038O1M (Deer, Male, GMU 038, Public Land Muzzle-loader) still available. If you move fast you could get one.
For those who don't understand "lists," those only really apply if you want more than one tag. There's a table on Page 19 that walks through this, but basically if you want two tags, you can get an A and a B, or a B and a B. You can't have two A tags.
Speaking of preference points there's a little confusing detail here that's worth knowing. You're allowed to put a special code (e.g. D-P-999-99-P for deer) as your "first choice" in the draw. If you do, you're guaranteed a preference point, AND you can still add more codes, AND also mark you'd like to be entered in the leftover draw. This does reduce your chances over just making your first choice your first pick, but if you plan to hunt in units with high quotas/draw percentages you can often get a preference point AND draw an A-list tag AND buy a leftover B list tag.
I did that this year, so I now have the trifecta: a preference point, an A-list tag, and a B-list tag. My tags are for lower-chance, higher-pressure zones (but still good ones), but I can still hunt two zones and save points for better draws in the future.