Looking on the internet deeply has found these results:

in early June is the most popular phrase on the web.

on early June 1106,000 results on the web	in early June 17,300,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
It doesn't look like this phrase is very popular! The search could not find any examples on the internet.	We were in early June for a week in Fonte Riccione. I wanted to in early June. We had 6 nights in early June, on the premises. Maybe in early June, someone was pushed out of a car into the Angeles Crest. We have been here in early June. The chapel was consecrated in early June 1900. The exclusive and beautiful Perini Navi sailor was in early June 2014 flagged in Malta and handed over to the owner. The EP Delegation visited the Korean Peninsula in early June 2010.

on early June

1106,000 results on the web

in early June

17,300,000 results on the web

More popular!

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

We were in early June for a week in Fonte Riccione.
I wanted to in early June.
We had 6 nights in early June, on the premises.
Maybe in early June, someone was pushed out of a car into the Angeles Crest.
We have been here in early June.
The chapel was consecrated in early June 1900.
The exclusive and beautiful Perini Navi sailor was in early June 2014 flagged in Malta and handed over to the owner.
The EP Delegation visited the Korean Peninsula in early June 2010.