Suppose there's a hypothetical scenario where there's only one head covering available (e.g. a baseball hat) — should I wear it or should my wife?
Whose head covering would take precedence and why?
The answer to this question can have various outcomes all depending on what you hold regarding kippa and what you hold regarding a woman's head covering. There are shittos which hold that wearing a kippa saves your from the issur min HaTorah of bechukasaim lo teilachu (Taz opinion in Orach Chaim 8:3). Then there are those who hold you can even learn without a head covering (Shu"t Maharshal #72). Then with regard to a woman's head covering you have many opinions who hold that all of the hair must be covered even in one's house, and then you have opinions which are melamed zechus that covering hair today is not required (Rav Yosef Messas Otzar Hamichtavim 3:1884, translated here, and Mayim Chaim 2:110). As you can see every scenario can change the halacha and based off which opinions you hold by. There is such a shitta that in your own home covering hair is not required even if there is a guest (check Nosei Keilem in Even Ha'ezer 115).
However, the more accepted approach according to most poskim is that wearing a kippa is only a minhag, and covering hair is surely more required. So simply put the wife would get the covering since that is more important. I do believe that such a case is highly unlikely to ever happen because one can use many items to cover ones head, but hypothetically the poskim would be more concerned about the woman's hair being uncovered.