Congrats on your new rescue dog!
Yes indeed hounds in general can be difficult to convince that they really do have to abide by housebreaking rules!
I'd make 100% sure he has no health issues such as intestinal parasites or urinary tract infections etc. Those can slow down housebreaking and make it impossible for a pup to do what you want.
Hopefully you have him on a schedule for eating, potty trips outside, and he's getting plenty of exercise to help him pass stool and urine while outside with you so you can praise him for correct behavior.
Make sure to clean up inside with an enzyme based cleaner to remove all odor. Nature's Miracle, Simple Solution, Odor Mute are some examples of those. You have to destroy odor so he can't find it with his super dog nose!
When you clean up pick up some of what he does to put outside - you can rinse paper towels of urine even to scent mark the outside as 'this is the spot'.
He may not understand not to go in his own space but hopefully he will not go in your space so tethering him to you with a 6 foot leash while he is in the house with you may work. This will let you rush him out if he appears to need to go or stop him in the process.
You didn't mention what food you feed and that can play a role too. A highly digestible food fed twice a day and picked up after 20 minutes can help control when he needs to go.
Leash walk him and give him time to mark and potty. If he can't go while on leash try a long line for potty area use so he can get far enough away from you to go.
Consider teaching him to ring a bell as described here
http://www.paw-rescue.org/PAW/PETTIPS/DogTip_TeachingToRingYou.php
http://www.clickersolutions.com/articles/2001b/bell.htm
to say he needs to go out.
Reward him for going outside. If he's not much on food rewards bring along a squeaky toy to play with him or something he really likes. It could even be once he potties he gets off leash run time in a safely enclosed area.
And if urine is the bigger issue try a belly band
http://www.pekeatzurescue.com/bellybands.htm
some people use a diaper around the 'waist' instead.
You can read more here
http://www.yolo.com/~pamgreen/adult_housebreak.html
I'd also suggest the nothing in life is free regimen which you can read about here
http://www.paws.org/cas/resources/fact_sheets_dogs/nothingfree.php
Hope this helps you!
None of this is anything new to me; most was part of the housebreaking book my vet recommened. Health concerns were already addressed and he eats Hills' Science Diet puppy food.
It is inconceivable to me that he will pee on his own pillow then walk over and pee by his crate. Plus he will poop then eat it. He seriously seems to be doing it to hide what he's done. He's done this 10 minutes after coming in from being out. You can tell he's messed by his body posture when you walk into the breezeway. He get plenty of attention as well. I've been told dogs don't do this out of defiance but it sure seems that way.
As you didn't include all that information I gave you there is no way to know what you do or do not know.
Science Diet is not considered a very high quality food except by those who sell it.
The ingredients in that could easily explain why he is eating his stool and messing frequently. You might try an actual high quality digestible food with him and see if that improves things. Some examples Blue Buffalo, California Natural, Wellness, Innova, Canidae, Eagle and others found in specialty pet supply stores could be a better choice than corn filled Science Diet.
I see you added a lot of info after I replied which I did not see at the time I replied to you.
Punishing him after the fact will not do as much good as praising him for correct behavior will. Dogs sometimes go due to stress and anxiety, or because they are marking territory but its not vindictive.
If you have not checked a stool sample for parasites lately it would be a good idea to do it again.
He is not going out frequently enough and he needs more than one walk a day. This is a young active breed of dog that typically needs lots of exercise.
Being out is not the same as being taken out and walked. The yard is pretty boring for a dog after a while and pottying on a walk is usually considered prefarable.
He might be marking his space and so the nothing in life is free program may help with that.
The belly band may help to keep him from being able to mark inside which can reduce the behavior particularly when he both gets to leave his scent outside and gets praised for doing so.
Dog Expert:Rescue, Train,Breed,Care
30+ yrs dog home vet care & nursing, rescue, behavior&training, responsible show breeding, genetics