Have you considered a shasta daisy. (Or is that yellow center a problem) Typically a fairly long bloom time that could begin in mid June and last close to a month depending on variety. One that works well for me is Snow Cap. Good strong stems keep it nicely upright at about 12 to 16 inches.

A David Phlox? Or would that be too tall?

I first thought of David Phlox also. So I checked my garden pictures to see when it would be in bloom. It seems that it doesn't start blooming till early to mid July. A great looking clean white and long lasting bloom to be sure, but I think it would be a little late for your purpose. Maybe in your zone, it would be a little earlier. It is worth checking that out.

Maybe perennial baby's breath? White obedient plant might work, be sure to get the one that doesn't spread so much. I think it is called Miss Manners. I got that one years ago, & wish it would spread a bit more. Hardy hibiscus, or a white clematis to climb over your hydrangea?

My Campanula persicifolia 'Alba' (white Peach-Leaf bellflower) blooms just about that time, the start of July.
CMK

Echinacea 'Pow Wow White' is a drought-tolerant, full sun, long blooming 2 ft. tall perennial with 3-4" blooms.

Shastas start to bloom in early July in my Z5 garden in northern MI. David phlox would be a few weeks later... Not sure how you would need to adjust the est. timing for your location.

There are some double shasta daisies where the yellow center would be less pronounced, although I do love shasta Becky for its habit, and for being a blooming machine.

Phlox David is also a great plant.

Beth
Z5 northern MI

If you like the idea of white phlox, but think 'David' might bloom too late for you, there is a rare found one called 'Midsummer White' that supposedly starts blooming earlier than any other white Phlox paniculata and is impervious to mildew as well as highly fragrant. Perennial Pleasures in Vermont sells it. The owner states that it starts blooming by the end of June in northern Vermont.

I have a few ordered for delivery at the end of this month, so hopefully it will be a good one! :-)

In case you are still searching/undecided....My shasta daisy 'snow cap' according to several pictures I have taken over 3 summers were beginning to bloom in the third week of June, at full bloom by July 1 and still attractive around July 10th. My shasta 'silver princess' blooms about 10 days later than the snow cap. You are a zone warmer, so bloom time may be a little earlier.