Clubs are obliged to pay for the month after the contract ends (i.e. July) if the player doesn't sign for another club.
There's nothing in the FA's rules to say this. The only circumstance where this has to happen is the one I mentioned above.
The appropriate part of the rules:
(iii) Within 7 days of the first Saturday in May, or the date of the last
competitive Match of the Club’s first team, whichever is the later, the Club
must give notice in writing to the Player indicating that either the Club
offers a re-engagement or, if appropriate, exercises any option contained
in the agreement.
(iv) If the notice offers re-engagement it must specify the period which
the Club is prepared to agree and the terms and conditions to apply,
which must be the same or not less favourable overall than those which
applied during the initial period of employment – or the option period (if
applicable).
(v) The Player must notify in writing the Club holding the registration within
28 days of receipt of the said notice whether or not the offer of re-
engagement is accepted.
(vi) If the offer is rejected the Player is immediately free to negotiate with
another Club.
(vii) If the Player does not reply in writing to the offer of re-engagement then
at the expiry of a period of 28 days, the Player is free to negotiate with
another Club.
(viii) In either of the instances as set out in Rules C1(j)(vi) and (vii) above,
the Club holding the Player’s registration has the right to receive
compensation. The Player’s registration for the new Club will not be
accepted until such time as the Club has confirmed in writing to The
Association that it will negotiate a compensation fee with the former Club
failing which it will abide by any decision taken by an appeal committee
comprising those persons pursuant to Rule C1(j)(xii) (a “League Appeals
Committeeâ€).
(ix) An existing agreement shall continue and have full force and effect
between the parties on the terms and conditions as those pertaining
prior to the expiration of the initial period of employment, or the option
period if appropriate, and the Player will continue to train and play as
instructed by the Club until 31st August, or the date on which the Player
registers for another Club, whichever is the sooner.