I believe you're reading that wrong. The eligibility period for this year (2016) ran from 6/1/15 - 5/31/16. It's premiere was well before the deadline, putting it into this eligiblity year. Episodes 7 thru 10 ran outside the eligibility period, but because of the hanging episode rule, they are also grouped into this year's eligiblity, because they aired before the return of the first round balloting (which technically was due at 10PM on 6/26/16, coincidentally right when the season finale ended).

This is all covered in the most current rules form, under 14c:

14c. Hanging Episodes for series eligibility: If an ongoing series has enough episodes in the current eligibility year to qualify as a series and has one or more episodes that are part of the series season, included the season/series finale, that fall into the subsequent eligibility year, the "hanging episodes" that are in a contiguous rollout on the same distribution platform join in eligibility the already-qualified-as-eligible episodes of the series, as long as the hanging episodes air prior to the return of the first round ballots. For example, a comedy series that regularly airs on a weekly basis that has six episodes in the 2016 eligibility year and has two more episodes of its series season airing subsequently and on the same platform in the 2017 eligibility year would enter all eight episodes in 2016 eligibility. On the other hand, a comedy series that has two episodes in the 2016 eligibility year and six more episodes airing subsequently on the same platform in the 2017 eligibility year would enter the eight episodes only in 2017. They would not be allowed to "sneak" the six episodes, e.g., post them on the internet or air them in an obscure time spot, in order to qualify the series in 2017.

For next year's eligibility, what is relevant is rule 14b. Specifically, shortening future seasons is acceptable, as long as they still are 6 episodes in length or longer:

14b. Shortened seasons for regular series: Once a show is established as a series, if additional shortened seasons are added to the series order (fewer than six episodes), it would still be considered a series, unless it was limited to one or two episodes (which would categorize it as a movie and a limited series respectively).

What really matters is the air date of next year's premiere. If it's June 1st or later, they'll miss out on the 2017 eligibility year, and roll forward right into the 2018 period. If however, the Season 7 premiere is May 31st, 2017 or earlier, it gets a lot less unclear, because I don't think they can meet the eligibility requirements for either 2017 (I think you need at LEAST 6 in a given year before counting "hanging episodes"), nor would they meet the requirements for 2018 (since it premiered in the 2017 year). I'd imagine for these reasons, combined with the length of production, that we see it air June or later next year.