New dogs coming into the household are never allowed together right away. They are separated initially by either crating or baby gates or something along those lines.
I don't even allow them in the same room with the new, crated dog the first night - they know there is a new dog here - they can hear it moving around in the crate, but nobody goes up to the crate. I want the new dog to relax and not feel cornered by unknown dogs. I want my dogs to relax and not feel they need to confront a newcomer.
Many times people think that because there is no fight, just lots of sniffing, etc, there is no stress. But it's incredibly stressful for most new dogs to be thrust into an established household - and it's stressful for the established group as well.
I know there are many times when two dogs meet and are getting along famously instantly - but I think this is the exception, not the rule - and you can go a long way toward easing the tension if they get to know eachother slowly.
Much depends on the layout of your house, but in my house, I'm able to have a new dog in the dining area, barred from the kitchen, and my house dogs barred from the kitchen but in the living room. They can see eachother, but no one gets near eachother. I would keep the new dog on leash and connected to me at first to make sure they aren't go to jump a gate. My dogs do not do that at this point.
If I want my dogs in the kitchen, I crate the new dog in the dining area, but leave the barrier up - so they can hear/smell eachother, but not get in eachother's space. If want to give the new dog more space to move around with me, i.e., let them in the kitchen, I move my dogs into the bedrooms or put them in the yard. And then shuffle everyone around again when it's time to rearrange it.
The new dog spends a lot of time crated the first couple of days - and gets one on one time with me inside and in the yard. I don't even usually take the new dog on a walk outside the yard for a week.
On the second or third day, depends on whether they are all calming down, I will move the gates, so they are now closer, but still not nose to nose. I might bring one of my dogs in the yard to meet the new dog. New dog on leash for this, perhaps both dogs. By this point, I have a sense of what might happen based on how they behaved just seeing eachother across the divides in the house.
But for those first couple of days no loose, nose to nose contact even over baby gates - I use them as a sort of airlock.
By day three or so, I may let one dog into the room with the crated new dog - they are usually calming down by now - used to the sight of eachother.
By day 3 or four I will allow one dog to meet the new dog with a baby gate between them - this has usually been uneventful. If those goes well, I may let one dog in with the new dog in the house. I introduce the new dog to each of my dogs individually first - then go to twos.
When I get to multiple dogs meeting the new dog, I move it back outside. You can see how this goes.
By day 4 or 5, everyone has met - although I save the introductions to my toy dogs for last - I want a good sense of this newcomer before I go there. I won't let the newcomer run with all of my dogs (6 now, but would have been 5 when latest came in) for a full week or two - and certainly not off leash with the toys until I have a good grip on the situation.
When dogs meet slowly over time, they get a sense of eachother, feel less challenged, there is less tension and likelihood of eruptions over things. That is not to say that you still could not end up with dogs that are not compatible, that could certainly happen no matter how long you make your introductions. But sometimes introductions made when everyeone is tense and adrenalin is high will be less pleasant than letting them all calm down and get used to eachother before meeting.
Even though they are not going nose to nose, they are gleaning a lot of information about eachother through how they move, smell, act, etc. So, they are likely less worried when they finally come nose to nose.
I've made the mistake of throwing two dogs together too quickly and ended up with a serious problem, so I don't do it anymore.
You may have had this problem with our without slow introductions - if the corgi is food possessive, than that is its own problem - but it can't be helped by putting a group of dogs that are unknown to eachother under stressful conditions and letting them make decisions about what happens. The Corgi should not be allowed to bully the hound mix, period - he can't corner her if he's tied to you with a leash. He shouldn't get the freedom to do these things until you've put some training on him and know him better. All of which can take awhile.
It can be a serious inconvenience to run a large pack safely and those first few days are critical and stress filled. Take the pressure off your little hound and structure the corgi. If he's targeting her for reasons other than food possession, you will have to evaluate if they can live safely together. He may be a bully, i.e., targeting a less dominant dog. I had a cattledog here once guarding me the first day he arrived - didn't want my Rottweiler near me - because of how much dog he was, I opted to rehome. It didn't help that his foster home had 8 cattledogs and 2 Rottweilers and the Rottweilers beat the tar out of him. I think he was partially guarding me and partially had developed a distrust of Rottweilers. It's possible we could have worked it out, but I didn't want my gentle Rottweiler mix to get hurt on my learning curve. Nor did I want her to be afraid in her home.