I'm looking for the term for a a long, unique intro in a tune. It was prevalent in the mid-20th century in musicals and the like.
In the context of mid-20th-century musicals, it's usually called the verse. This meaning of "verse" was typical for song written for Broadway.
The usage is informal. But it's explained well, for example, in a post from Jazz Guitar Online.
The terms Verse and Chorus meant different things in the Broadway tradition to how we now commonly use them. The verse is the preliminary musical introduction to bridge the spoken dialog of the play to the song. Melodically it was often more plain to match a musical speaking tradition that was used in opera with the recitative. The main body of the tune (often AABA or ABAC in form) made up what was called the chorus. Within jazz terminology this became known as the head. The head is distinct from the solo where the chord changes of the chorus are used to improvise over. In opera the Chorus would be the Aria. Within the chorus you may have a bridge if the structure is AABA. Verses fell out of favor as recordings of these songs became popular. Because the verses were a device to move from spoken dialog to singing within a play they were often not included in recordings by artists of the day. When Hollywood began making movie musicals the verse fell out of favor as well. Camera tricks and instant scene changes not possible for stage productions rendered the verse mostly obsolete. (SOURCE: "What does the 'verse' mean?", specifically post #4 by setemupjoe at 05-28-2019, 11:09 AM.)
Generally speaking, modern songs (pop if you like) are split into verses and chosuses, occasionallyy a 'middle eight' or bridge section, but mainly verses and choruses - the verses differring word-wise, but the choruses, in the main, being the repetitive parts.
Not certain why, but the pre-amble of older songs is referred to as the verse. It often seemed to have little to do with the song itself - so much so that in general renditions it's rarely sung. It was often sung rubato - as if to tease the listeners to 'name that song'. Very much like the intros to musicals!
I recently performed 'My Funny Valentine' - with verse . That verse pretty well took the Mickey out of the singers inamorato, opposed to the rest of the song's words.
I suppose the problem then is - what is the next part of the song called, after the 'verse' and before the chorus..?
I would suggest the term "prelude". A prelude is generally a separate piece of music from whatever comes after it, but many preludes are attached to the next piece and may as well be a long intro.