If you want a botanical challenge, June is the month you can start collecting Sedge seeds and identifying them using a good field guide, such as Colour Identification Guide to the Grasses, Sedges, Rushes and Ferns of the British Isles and north-western Europe by Francis Rose. Many sedges are wetland species or plants of waterlogged places, and June is a good month to visit the étangs in the Foret de Preuilly to see many other plants at their best and in flower.
The very gregarious Marbled White butterfly will be flying on sunny calcareous grasslands, taking advantage of all the flowers and the warm conditions. Where you see one, you will see several. The caterpillars feed on several species of grass and this is one of the most common butterflies in France.
June is the month you can see the Hairstreak butterflies that are associated with oak. They have a very brief flight period, and spend a lot of time high up in the oak trees. However if you are in the right place at the right time they can be seen nectaring at bramble flowers on the forest fringe or drinking from a puddle near a large Holm Oak in the grounds of the Chateau of Chenonceau.
It's not just butterflies that will catch your attention in June either. Some of the day flying moths, such as the Burnets, are gorgeous. Once again, mostly to be seen on the calcareous grasslands. They are particularly attracted to flowers like brambles and scabious.
Everywhere you look in June there will be something of interest in the forests and prairies.