If a +1/+1 counter hasn't been placed yet at the moment an end step begins, Fairgrounds Trumpeter's ability doesn't trigger at all. If another ability triggers during the end step and puts a +1/+1 counter on a permanent you control, you won't put an additional +1/+1 counter on Fairgrounds Trumpeter.

Fairgrounds Trumpeter's ability triggers if, at any point during this turn, a +1/+1 counter was placed on a permanent that you controlled as the counter was placed. It doesn't matter whether you still control the permanent or whether it still has a counter.