It is almost certainly just a really really bad idea and you should stick to either one. At novel length it has been made to work by people like Charles Dickens and Cormac McCarthy, but at short story length I can't imagine it being successful. Novels can be episodic in structure, but a short story is almost always a unity.
Furthermore, I would challenge this notion that first person allows you to get into the head of a character. On the contrary, it is the POV that most shuts the reader out of the character's head. An omniscient narrator, after all, is omniscient. They can go anywhere they like.
A close third person narrator, similarly, can go anywhere they like while focussing on just on character. (Close third, after all, is just omniscience with a stalker complex.)
But first person is always, in some sense or another, the voice of the character. It is what that character chooses to say, what they choose to reveal. But a character who chooses to reveal may also choose to conceal. In fact, this is what we assume tacitly about everyone we meet and listen to. They are concealing things. And what the conceal, though we can only guess at it, is just a revealing as what they show.
If has been well said that a first person narrator is always an unreliable narrator. We are always in a position to assume that what the character is choosing to tell us is something other than the truth, and never in a position to be sure that it is the truth. We may catch them in a lie, but we can never catch them in perfect verity.
This is the genius of the first person narrative when it is done well (which it very seldom is). It is like meeting a person in real life. We are never entirely sure that they are not hiding something from us, and that creates a tension in the relationship that can never wholly be removed.
So if the desire is to get the reader into the head of a character and know what is going on there, then third person is the right choice. If you want to create a mystery about what is going on there, a tension between what they tell us they are thinking and what they might actually be thinking, then first is the tool for the job.
But in a short story, pick one and stick with it.