I just checked the official website,http://www.toureiffel.paris/en,
& tickets for July 3rd will be available for purchase on Monday, April 4th at 8:30AM Paris time.
So keep checking for your desired date of July 4th, & tickets should be available for booking on Tuesday, April 5th, at 8:30AM Paris time.
When I was ready to purchase our ET tickets, a couple of weeks ago (for June),
I had 2 tabs open on my computer screen, one for tickets to the summit & one for tickets to the 2nd level.
I even called my bank to notify them of my purchase, just in case, as I didn't want my bank to deny my "France" purchase.
Make sure you select the date, the type of tickets you wish, the number of tickets, etc... & be ready to click on the "Confirm" bar, right at 8:30AM Paris time.
Don't be surprised if tickets later than 5:30PM aren't available.
Paris will be hosting the Euro Cup from June 10 -- July 10th, so I presume the tour companies have snatched all the tickets later than 17:30 (5:30PM)
Good luck!