In formal writing, esp., in the fields of programming, linguistics, the word non-final is already in use.
On Stackoverflow (java):
What is (a) non-final variable?
In René Kager, A Metrical Theory of Stress and Destressing in English and Dutch (Lingistics) -- Google Books:
… if the rightmost stress is on the final syllable, the primary stress can either be on the final syllable (112a), or on the rightmost stressed non-final syllable (112b) … [emphasis added]
See also ***** :
medial on ODO
1.2 Phonetics (Of a speech sound) in the middle of a word:
A total of 15 items were included on this task requiring students to delete initial, medial, and final phonemes from words [emphasis added]
Although medial, tends to suggest something closer to the middle, it's not necessarily so. In the example above, anything that is not initial or final is referred to as medial.
***** However, medial excludes the initial, which is why it is not a candidate for the OP's case.