June is a season of hitoe. As for obi, Hakata obi which has no back or summer obi which is transparent might be good.
If you don’t have them, obi of thin awase is good.
Formally small item such as obiage and obisime are also for summer. But only small part of obiage is visible, and some people use the thin one for winter. Some obisime can be used throughout a year, and it is economical to select this kind of one.
unless you go to formal place, it is a good idea to use hanhabaobi.
There are various ways of tying of obi, and some are introduced in book or website. If you find your favorite one, you should challenge it.
June is a month when we have much rain. It is hard to wear rain prevention clothes such as coat and zouri for rain if you are not used to them. Therefore we must consider a measure to “have kimono made of waterproof material” and “to have extra tabi to swtich if feet get wet”.
In June, willow and hydrangea are used as design often, but in latter half of this month, people tend to like firefly and arrow bamboo which are more summerlike.