Good post HuntsmanTough question to answer, but I will give my thoughts and then maybe more others will chime in also. I have no set formula. I now go many years between full outcrosses. I did more outcrosses 30 years ago when I was trying to gather the bloodlines I admired so I could carry them forward into the future. Mostly these were lines I had seen while judging SPO trials, which is a great way to acquire knowledge about what you want (and don't want) in your own kennel. After I had the lines I wanted in the kennel, I went outside less and less, and seldom go out today. Everyone's situation is different, so you may already have what you need right from the start. But you should never be totally satisfied - there is always something you can improve in your hounds.
I prefer now to work with a partial outcross, meaning the hound I am breeding to is related to mine, but has some lines in his/her pedigree that are different from mine. Outcrosses give you hybrid vigor, so you may get a super hound in that first generation after the cross, but outbreeding also scatters the genetics you already have, so they need to be done carefully and only after much research into the lines you will be introducing into yours. Outcrosses can also be failures and have to be discarded. And I agree with Will, sometimes it can take years to get it out of your kennels. (Been there and done that!)
Speaking of research, if you find the owner of the outcross hounds you want to use will not openly discuss with you the positives and negatives of his/her line, you may want to look elsewhere. There is nothing worse than the owner who hides information, perhaps for his/her own gain, and lets you find out later about a serious problem you have injected into your lines. An example might be something like not mentioning the hound you want to use will sometimes have epileptic seizures, and the dog's grandsire had them too. Maybe that's a poor example, so it could be just that his line has poor feet, or bleeds heavily when hunted, or is known to die by age 10, or sometimes produces bad bites, cherry eyes, etc, etc. Better to deal with honest, open people you can trust to tell you what you need to know. No hound or bloodline is perfect and without weaknesses in some way. As Will said above, dig, dig.
The more satisfied you become with your own lines, the more reluctant you will become to add new lines. It may also get to be harder and harder to find other lines that interest you, or that you think will add something positive to your lines. When you do find an outcross that works, you may decide to add more of that bloodline later, but the second and third time it's not really an outcross any more, is it?
I think the time to outcross is when you decide there is something lacking in your hounds that you cannot get from within. For example, some of my biggest outcrosses have been to show champions to add conformation to my hounds. I knew I could get conformation quickly that way, as opposed to breeding and selecting for it using only my own lines. We don't have the lifespans of Methuselah, so sometimes an outcross is preferable to selecting for a trait over time. You might decide you want more size in your hounds, or stronger noses, or more hunt/search, or whatever, and maybe the best way to get it is to crossbreed to a GREAT INDIVIDUAL from a FAMILY that is known to be strong in the trait your hound lines might be needing. If you know that individual is a strong producer of that trait, all the better.
Just remember the outcross will likely bring other traits too, so some culling of the offspring may need to happen. If you can't cull, you should not try your hand at breeding, because sooner or later you will need to cull hounds from your program. But culling does not have to mean you are putting hounds down. Often a cull to you will be a brag hound for someone else. I used to smile when I would hear John New refer to FC. Indian Hills Majer as a "cull dog", knowing how the rest of us felt about Majer (a dog John was breeder of). The Walker brothers, when developing their famous Kentucky foxhound lines, were known to cull severely, in ways most of us would not have a heart to do today. Times were different then, but that does not mean culling is no longer important to a breeder.
Outcrossing should make you a little nervous. It takes courage. Sometimes you might rather let someone else try it first.
The only thing I can add is
Know exactly what you expect from your out cross. As mentioned dig dig and dig some more but keep in mined don’t go off anyone’s advice without seeing first hand how the dogs offspring turn out is he/ she putting the Traits your looking for in there pups. Iv drove meany miles over the years to go watch pups run from a dog I was interested in making an out cross with
I especially liked your statement about making partial out crosses That’s how I always tried to make an out cross IMO it’s safer Not as Likely to get those hidden bad traits back behind the dog that occasionally rises from the dead
Once agin good post Tim