Actually, Jan, your flight departs at 00.00 UTC and the offset flight departs 23.59 UTC. You can ser this if you aveces your airline's flight schedule without logging in to server (without logging in, all displayed times are UTC).
There are two possibilities usually happening, one is the slot window is overbooked, and second one is slot is there but not showing.
In the first possibility, if you switch flight plan, there is nothing that can be done as simply there are e.g. 11 slots on a 10 slot airport in that particular slot window.if you decide to not move schedule, you can actually count additional slot in that particular slot window when you do manual count on airport's slot page.
The second possibility is the better case scenario, that the shoudl-be-free slot is not showing up, in such case slot refresh can help.
Of you click a flight affected with slot issue, and you do not click to save it but rather click away, that flight and slot on the original aircraft is not affected.