You have 2 unicorn covering sessions a day, of 1 minute each. You can use them any moment you want.
With 'day' I mean the time between two Howrse updates, I don't know at what time they take place in your time zone. You can see that if you go to the page of one of the horses you have bedded today. It will say: Your horse will wake up in [...] hours. That is the update time.

That means: if you wan to cover more unicorns, you need to have them all ready (stud chosen, put on the BMI you want to use), each in a different browser tab, just don't click the 'cover my mare' button yet.
As soon as you have all the unicorns you want to cover ready, you start the covering session by clicking the 'cover my mare' button on the first uni, and then go to all the other tabs and cover as many as you can within that minute.
That was your first covering session of that day.

You can do one more if you want, in the same way as described.

The info about unicorn covering sessions can be found in the Breeder's Manual

Yeah I get that - but my last unicorn covering was yesterday morning (before the recent update). Then I went to cover my unicorns this afternoon, and it says that I only have 1 covering left (now none), and my next available covering isn't until tonight at 11pm. When my previous covering was yesterday morning, should I not have 2 coverings at this point?

Which mares are you trying to cover? I see quite a few that can't be covered because they're on the 10 hour cooldown between covers.

I've not heard of anyone else with this issue, although I suppose something else could have bugged out with everything else going on.

I just covered 6 of them, but I was planning on doing another round of coverings later tonight for my team, and it says I have 0/2 Unicorn Coverings left today.

I honestly figure its from all the stuff thats going on - mostly wanted to see if anyone else was having the issue. Its not the end of the world lol