look into Indian reservation hunts, in SD, tags are on a first come first go, but if you apply when they open for sale, your pretty sure you'll get one on day one!
you then can also TRY and draw a state tag, and make the trip into a two buck(deer) hunt, many reservations you can add a pronghorn as well,

I've done a few reservation hunts and did them on a DIY deal, as I Have done ALL my deer hunts in the USA!
I am guessing you want Rifle hunts too!

as most states out from you are Over the counter on bow tags for whitetails
ND/SD, have a TON of public lands too, and many deer too boot
have hunted both and done very well every trip
33 hrs of driving for me however being here in PA , LOL