Three Roosters?

They will not divide up they way you think they should: ten for you, ten for me, ten for him. There will be squabbles and fights, some serious, and some possibly deadly. A cockerel has to pick his flock from among the hens already "belonging" to a senior rooster, and the senior will not give them up willingly. The best way to avoid this is to physically separate the flocks with good fencing and housing, or to have enough acreage that they can separate into "tribes" as @Shadrach's did. But @Ridgerunner will tell you that with living animals you don't get guarantees, they are individuals and anything is possible. I had about 25 hens and raised up a very good cockerel as subordinate to my flockmaster. They mostly free ranged. It seemed to be working out very well. Until the morning I came out to find the cockerel dead. You just never know.
 
I am wanting a better ratio as well as a lower chance of inbreeding if that makes sense.
I have no idea what you mean by "better" ratio. There are pros and cons to this so "better" just depends on which of those you look at.

A lower chance of inbreeding is logical, depending on how you manage that. There are different techniques to control inbreeding. Having more chickens and randomly saving replacements does help with that. But there are other techniques.

The yard has an uneven amount of yardage. My father built it. The main yard part is 16’ x 8’ the extension part is more like 20’ x 8’ we also have logs and old tires in it as well for them to jump on and be entertained with and to find bugs underneath.
So 38 chickens and 288 square feet. I personally do not believe in the magic numbers you often see for square footage chicken have to have. That's still tight. You are not guaranteed to have problems if you add those 12 or 13 but problems are more likely. The more you crowd them the more likely you are to have behavioral issues, you have to work harder, and you have less flexibility to handle issues that pop up. Crowding them like that it is possible (if not likely) your chicken keeping experience will be less enjoyable.

I understand you plan to let them roam some this spring and summer. How about Fall and Winter? When they cannot roam they will have to get along in that tight space. You often need more room to integrate than you need after they are all mature and integrated.

You say you want to avoid inbreeding. That means you will be hatching chicks. How many do you plan to wind up with? It sounds like it could be a lot more than 38. What do you plan to do with those chicks?

So would 3 roos be the best option?
I haven't read anything that makes me think adding 12 more girls is a good idea, let alone another male. You can try it and you might get it to work out but I would not.
 
So what I am hearing is no 3rd roo. Got it! My parents want to breed our hens so that we don’t have to buy chicks in the future. Chicks this year were really expensive so breeding our own is our plan.
We are keeping our roos with our hens full time. I will not get a third roo. Thank you all for your help!
 

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