At or In a particular time of need?

Question

When should I use in or at for a particular point in time?

Example:

1. Joe and Jill couldn't reach us at our time of need. They reached us too late.
2. Big Corporation reaches customers at their time of need.
3. The ambulance reached me precisely at my time of need.

Please note that in each of these situations, the point in time is a brief period.

I recall the general rule of thumb as:

in - periods of time. Ex. "I'll see you in June.", "I'll see you in five minutes."
on - days and dates. Ex. "I'll see you on Tuesday."
at - particular time. Ex. "I'll see you at 6 o'clock."

But I'm unsure which is correct in the above examples.

Which is the correct preposition in each of the above examples?

Answer 1

To me, the structure of all the sentences look somewhat strange. On the other hand, you are quite right about those rules mentioned. However, we cannot imply them to all the instances.

"Thank god, you reached on time!" or "The train is on time" - I'm talking about particular time; no days/dates required as your rule says!

Now coming to those sentences in concern.

The word that makes all the sentences strange there is need + using a preposition. They all can be written without using prepositions and I think they'll look natural then. The adverb when takes care of that!

Joe and Jill could not reach when we needed them. They reached late.
Good companies reach their customers when they need them.
The ambulance reached when I needed it.

If you want to emphasize the exact time, you may simply use it that way with preposition in.

Joe and Jill could not reach in (exact) time when we needed them. They reached late.
Good companies reach their customers in (exact) time when they need them.
The ambulance reached in (exact) time when I needed it.