Fungi season - when to go?

Hi everyone, warm welcome from cold Poland. We are around 0C now 
I'm preparing to visit Australia first time and complete the Overland Track. Now, I take my time do my "homework" and search for some information on the Internet, however I would like to ask for some help to dispel my doubts.
Short question from me: when should I book the entrance to admire OT fungi season and all these colorful autumn views?
I'd like to take advantage of 'fungi season' which looks absolutely unique. If I'm right, April is the best period (if I'm wrong, please correct me)? But when exactly would you suggest to come? First half of the month or maybe in the middle or later?
It would be breat to visit Tasmania in combination nice weather conditions, temperature and fungi availability
Just want to be prepared with all organizational steps before booking the flights.
Thanks a lot,
B.

I'm preparing to visit Australia first time and complete the Overland Track. Now, I take my time do my "homework" and search for some information on the Internet, however I would like to ask for some help to dispel my doubts.
Short question from me: when should I book the entrance to admire OT fungi season and all these colorful autumn views?
I'd like to take advantage of 'fungi season' which looks absolutely unique. If I'm right, April is the best period (if I'm wrong, please correct me)? But when exactly would you suggest to come? First half of the month or maybe in the middle or later?
It would be breat to visit Tasmania in combination nice weather conditions, temperature and fungi availability

Just want to be prepared with all organizational steps before booking the flights.
Thanks a lot,
B.