The 60 vote requirement is needed to end a filibuster. The Senate, in, IIRC the mid 70's, cooked up this "procedural filibuster" bullchit. All this is is a group of Senators expressing their intent to filibuster, and whatever they are doing gets shelved so they can go on with other stuff.
It's hokey, and entirely consistent with the way the Senate has fallen apart since the 17A negated the whole purpose of the body.
But when do they have to declare their hokey filibuster? I assume before any vote begins. They cannot do it mid-vote. Right?