Every studio sets two limits. One is how far into the future its calendar opens — a few weeks for most, several months for studios whose clients rebook every visit. The other is how close to the appointment you can still book, which is usually somewhere between an hour and a full day. Both are the studio's settings, and both are simply reflected in the times you are offered.
So the calendar showing nothing beyond a certain date rarely means the studio is fully booked. It usually means the calendar has not opened that far yet. Booking opens on a rolling basis, so a date that is unavailable today may be available next week.
If you need a date beyond the window — a wedding, a holiday, anything fixed months out — message the studio and ask them to hold it. Studios do this routinely for appointments that matter, and it is far more reliable than waiting for the calendar to roll around to it.