Reservations at the Svenska Turistforeningen mountain huts

You don't typically book unless you are a large group, or it's a 'special' hut.

Many people decide their hiking based on the weather, and this cannot be predicted.

If there is a shortage of beds (unlikely, especially in June), there is always enough space and mattresses. (Children and elderly has priority for the beds.)

(In ancient Scandinavian tradition, those who have stayed one night must yield to new arrivers in remote wayside huts. This is still rule in the Norwegian mountain huts (though hardly ever an issue). I don't know if this rule applies as such to the Swedish ones, but you get the mentality.)

I quickly checked the Abiskojaure hut, which has "booking & price" information only in Swedish. Google translate should help you out, but the relevant portion from the info would be:

Booking: It's not possible to book a place in advance in the STF mountain huts. You are always guaranteed a bed. In special cases, when the beds are occupied, you will be offered a place on the floor.

Prices: If you're not a member of STF, add 100 SEK to the prices shown.

#2,
You don't have to pay that SEK 100 supplement if you are a member of any Hostelling International affiliated society, such as YHA in the UK, Danhostel in Denmark or DNT in Norway.

It's correct the huts aren't bookable. Except at the larger lodges (Swedish: fjällstationer) such as Abisko or Saltoluokta, where you can reserve both beds and rooms. It's possible (and quite normal) to arrive also to those lodges without a reservation. As said above, often, due to the changing weather, you may not be able to know the exact dates of arrival or departure. If it rains on a certain day you may want not to walk that day, etc.

It's also true it's not that likely that huts are full in June, particularly in that northerly located area since they have just opened after a very long winter and a spring with plenty of valley terrain resembling wetlands (normal). In late July or the first week(s) of August, however, it's not unlikely that the huts are more or less full.

Thanks all. It’s good to know that we don’t need to stick to a schedule; especially since our kids are with us and I really don’t feel like packing the four person tent. :)