My name-choosing was sort of a multi-step process, because I hated my birth name long before I even came out as trans. As a result, I decided on a whim to switch from my birth name to another girl-name (Diane) sometime arounbd middle school. There wasn't any special reason I chose Diane-- it was just a name I had randomly assigned to a roleplaying character who I got fairly attached to, so I adopted the name for myself. However, when I came out as trans after my freshman year of college, it was a lot harder to let go of a chosen name than it was to let go of a stupid name my parents had given me, so it took me like a year and a half to find a proper guy name for myself! Eventually, a friend pointed me towards "Deyan," which had essentialy the same consonants and intonation with different vowel sounds but was a decidedly male name, so that became my new name.
Of course, nobody can ever pronounce Deyan (I've gotten everything from Daniel to Dayla, depending on how people read my gender and how bad their hearing is), so I frequently just introduce myself as Crow, which is my favorite animal and appears elsewhere in my legal name.
There's nothing wrong with having multiple names you go by, so if you want to hold onto your birth name but introduce yourself as another name (which you could insert as a middle name) while you wait for your transition to progress, I don't think anyone will think anything of it-- lots of people are known by their first name to some people and their middle or last name to others.