Is there a word for "all but the first one"?

Is there a word that refers to all items in a series except the first one? Example:

The tests are successful only on the first run. On all subsequent runs, following the first one, the tests fail.

I was thinking of the word "penultimate" (the one before the last one) and was wondering if there is a similar word that could be used in the case described above.

You could use noninitial.

Noninitial: Not occurring at the beginning of a word, phrase, or sentence; not initial.

Example: Noninitial syllables/stress. [M-W]

In your example:

Initial tests were successful but noninitial tests failed.

As has been said in the comments, subsequent is the word.

The tests are successful only on the first run. On all subsequent runs, the tests fail.

In fact you could get away with:

The tests are successful on the first run. On all subsequent runs the tests fail.

Nobody could possibly be confused about what this means. There is no ambiguity.

In computer science, there is a kind of algorithm that works on lists, and makes exactly the distinction between the first and all other elements of a list.

head is the term for the first element.

tail is the term for the remaining list after removing the head.

rest could be an alternative to tail.

I'm not sure that works for your practical case, but it's related because the whole concept of lists is based on exactly the distinction you make.

The terms are used like this in the context of functional programming.
There are shell commands with the same names, working on lines of text files, which are related, but different.

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.