June may not sound like the best month for birding in Athens, since the weather is usually very hot and dry. However, it may be the best month to enjoy late comers, like Rufous Bush Robins and Black-headed Buntings.
Black-headed Buntings are coming in May and stay only until July. Their distribution near Athens is limited, yet there are still a few places holding several pairs.
Furthermore, all the other resident or migrant breeders (Ruppell's Warbler, Cretzschmar's Bunting, Rock Sparrow, etc) are still active, provided there's not any strong heat wave and you are not birding at noon!