I would use:
- Unfortunately, the next available visit-time is in the middle of June.
- Unfortunately, the next available visit-time isn't until mid-June.
- Unfortunately, there is no available visit-time until the middle of June.
I would also probably say appointment, or even just time, instead of visit-time, since it will likely be clear from context that the time is for visiting.
You are right about amidst and midst. The same goes for amid, which sounds less literary to me.
However, in the middle of is fine and much more common. Though it can be used in many of the same places as amid, amidst, and midst, it isn't limited to the situations you describe — it can also be used to describe location in a single unit of time or a single thing/place. For example, middle can be used in these sentences but the other words can't:
- They were sitting in the middle of the room.
- Drill a hole in the middle of that board.
- Alex was in the middle of final exams and wasn't sleeping well.
- The sun was shining in the middle of the afternoon when I got a knock on my door.
- Jordan woke up in the middle of a strange forest.
- Jordan woke up in the middle of a group of strange trees.
- Jordan woke up amidst strange trees in a dark forest.
- Literary or poetic: Jordan woke up amidst a forest.
- Jordan woke up amid a forest of pine trees.
- In the middle of the last song, the lip-sync track suddenly stopped playing.
- He's leaving in the middle of next month.
Mid-June is also perfectly acceptable.
