There are 3 or 4 specializations available for each class. Each specialization is focused on a particular playstyle, and will come with its own abilities. To answer your question, no there are no "hidden" specializations, and no more specializations are later unlocked.
Starting at level 15, you get to choose your Talents. Each "tier" has 3 available talents to choose from, and you may choose 1 of the 3 from a given tier. More tiers are further unlocked at levels 30, 45, 60, 75, 90, and 100. Most of the talents are the same regardless of your class specialization, however some can change depending on your specialization.
When you hit level 30, you get Dual Talent Specialization. This means that you can switch between two specializations depending on what you would prefer to do. You'll likely maintain separate gear for each specialization, since gear that is good for one specialization might not be good for another. Though for your Warlock, this is less of an issue. I would be more concerned with someone trying to tank or heal in gear designed for damage.
If you would like to try out a different specialization, you do have the option of respeccing:
Changing a character's talent build is called respeccing or is referred to as a respec. Characters can unlearn all of their talents (becoming untalented) at their respective class trainers, at a cost. Unlearning talents scales up the more times it is done: 1 gold, 5 gold, 10 gold, 15 gold, etc., up to 50 gold.
The cost to unlearn talents decays over time. This cost will be reduced by a rate of 5 gold per month to a minimum of 10 gold.
Finally, it's worth noting that the system can always change. It used to be that every class had only 3 specializations. Back in WotLK, the "specialization" was never a thing, and each class could freely assign talent points (51 points at level 80) across their 3 talent trees. It is always possible that Blizzard will change the system as time goes on.
