Step 1: Distinguish "the project" from "the very next activity".
The key word in this question is "succeeding activity", singular and immediate, not the entire project finish date, this hints strongly that we are dealing with a local, activity level measure of slack rather than a project wide one.
Step 2: Recall the two main float types.
Total float is the delay an activity can absorb without pushing back the whole project's completion, while free float is the delay an activity can absorb without disturbing the early start of the very next activity that depends on it, free float is always the smaller, more locally restrictive of the two.
Step 3: Match the definition to the wording.
Since the question asks specifically about not affecting the start of the succeeding activity, and not about the overall project deadline, this points directly to free float rather than total float.
Step 4: Conclude.
So the correct term is free float.
\[ \boxed{\text{Free float}} \]