second-chance exam for those who failed the first exam

When I went to school, we used the term makeup exam:

Oxford dictionary (American English):

North American  A supplementary test or assignment given to a student who missed or failed the original one:       ‘ Tony has a makeup exam’

Merriam-Webster's Learner’s Dictionary:

US : a special test for a student who has missed or failed a previous test
• He’ll have to pass the makeup to graduate.
— often used before another noun
• a makeup exam/test/quiz
— see also make up at make ↙

from make at Merriam-Webster's Learner’s Dictionary:

make up [phrasal verb]
• make up for (something) : to do or have something as a way of correcting or improving (something else)
• He wanted to make up for [=atone for] neglecting his children by spending more time with them.
• She tried to make up for lost time by working extra hard.
• What the movie lacks in plot it makes up for in special effects.

i.e., a makeup exam is offered to make up for the fact that a student missed the original exam (a way of correcting) or failed it (improving).

Note that both definitions indicate “North American” or “US”.

See also Is it makeup or make-up or make up?

Supplementary exams — JCU Australia

A supplementary exam is an additional exam (or other form of assessment) that may be approved for a student in the following circumstances:

• A student who has come close to passing a subject and meets the relevant College guidelines for awarding a supplementary exam.
• A student who has applied for special consideration due to unexpected circumstances impacting on their performance during the first exam.

This is a second chance to pass the subject and students must pass the supplementary exam to gain a pass overall. The best possible result from a supplementary exam is SP (supplementary pass) or SS (supplementary pass in an ungraded subject).

Supplementary exams are scheduled after the formal exam period, and may not be available for all subjects, or for all examination types.

In my experience in the U.S., we call these re-takes or repeat exams. You could also call it a do-over.

From Merriam-Webster:

re-take: to take or receive again

repeat: to make, do, or perform again

do-over: to do (something) again

Repeat exam probably works best with your example:

I proctored the exam, and I will also proctor the repeat exam.