While it is apparently not used as much as non-initial, Non-first is an option that is used technically and can be general-purpose:
In the data, we find speakers using HT for producing non-first list items only. (Time and Emergence in Grammar: Dislocation..., by Doehler, Stefani and Horlacher)
or
Also, Nokia’s big payday is now a non-first quarter affair. (techcrunch article)
and
Therefore, due to the different location in a channel gain period, the sensing slots could be classified into two categories: first slot and non-first slot, and they are defined as (6). (EURASIP Journal on Wireless Comm...)
The first run was successful, while non-first results varied.
Edit including suggestions (Thanks, @user253751), just to keep keywords for the search engines:
"Non-first quarter" should be parsed as "non-{first quarter}" rather than "{non-first} quarter", making this construction the odd-ball in the three preceding examples. Another example is "Non-first order coupling". Non-x can be ambiguous at times; one wonders if the "non" (negation) is distributed or the "non-first". Anyhow, in the "order couplings" case I think it's fairly clear that the order is: "{{non {first order}} coupling}, or in another notation: Non-{first order} coupling "First order" has a specific meaning which "non-" is trying to negate. That is, check whether the coupling is first-order, and conclude that it's not - rather than checking which order the coupling is, and concluding that the order isn't 1.