Of all the classes, it seems least likely that the cleric will get more subclasses. The feature gives them their proficiencies and not much else, and given that it represents their role in the church or religion (sort of, as I say it hasn't really been explored) there isn't much room to expand.
Or is there? The answer is yes by the way. Basically, there are two major archetypes of priest in fantasy role playing, the warrior priest and the more bookish congregation leader, but these do not paint a very good picture of priesthood as a whole. Rather, they represent a highly christo-centric view of that role, particularly as it developed in Europe. Throughout the world priesthood takes on many forms, which very rarely coincide directly with the ones presented here, and there are a number of examples of roles taken even in christianity that are left untapped. To this end I will present some examples here, drawn from my admittedly limited understanding.
A few notes before I begin: I will be talking primarily about cultures and religions alien to my own. If you belong to a religion or other group about which I talk here, and you feel I misrepresented your faith, please feel free to say so. That said, I am drawing on those faiths for inspiration for a fantasy game, rather than trying to produce an exact replica, so my primary aim is to present something that would be fun, at least when better polished. Second, I am leaving aside a number of traditions and faiths that might very readily be adapted to this, particularly a number of animist or shamanist belief systems as it seems likely that Paizo will give them their own distinct class. Without further ado, let's begin.
The missionary: this type of priesthood is one fairly specific to christianity (even though in many fantasy worlds many religions are evangelistic). It requires a very specific set of theological impetuses for a religion to want to go out and collect more followers. That said, a player might want to take on the role of a character who goes out into the world to bring knowledge of their faith to others to persuade them to join their religion. My mechanical thoughts are that your character should be given a proficiency in diplomacy automatically, and, each time they manage to persuade another character to take some action in line with their gods teachings they should gain some kind of blessing. I don't know exactly where the limits lie however, especially since it would depend very much on your game.
The religious lawyer: this type of priesthood is drawn from my understanding of priesthood in islam and judaism. My understanding is this: in belief systems with complex sets of strictures, edicts, commandments etc., one important role of the priesthood is to interpret the sacred law, to be able to make judgements, and to discuss that law, and also to put it into practice. My thoughts here are that there are two possible mechanical paths. Either way I think one should have some kind of legal lore proficiency, but the options are these: either, which was my first thought, you act primarily as interpreter, and have a feat which at some interval (say once a month) allows you to commit an act anathema to your deity and get away scot free, with a successful legal lore check to find a loophole. The second, which I like less, is as more of an enforcer, with a built in ability similar to the spell anathematic reprisal, to rebuke those who break the religious laws.
The final type I will cover here, though there are many others, is that of the ritualist priest. Tanking inspiration from a plethora of sources, though I am thinking particularly of the Japanese shinto priesthood, and also of traditions such as hinduism and confucianism, the role of the ritualist priest is less one of community leadership and more one of making sure the appropriate rituals are completed. In my conception of this role actual belief plays a relatively small part, though it is prerequisite. Rather, the priest job is simply, as above, to go through the appropriate motions and chants at the appointed times. Mechanically I imagine that, in addition to some specific lore on rituals, such a cleric would have on the one hand the requirement to stop their adventuring at certain predetermined(likely by a timetable composed by the player) hours of the day to perform rituals, but in response to that they would get some kind of blessing. This would again be dependent on the game for how well it worked, as it would require the GM to keep track of time, and for the player to keep track of the cycles whereby their precomposed timetable of rites would work. There would also need to be restrictions on said timetables of rites to prevent their being exploited.
Edit, as I clicked post too early: As you can see, these ideas lack mechanical polish, but I hope also that they demonstrated at least some of the possible variety and untapped potential of the cleric's doctrine. This list of ideas isn't even close to complete. Just on the christian end of things (which is where I am the most familiar), I might suggest the scholarly theologer, the monk (no, the other kind) who takes on a particularly strict set of tenets, and the pilgrim, who spends much of their time traveling between holy sites. Either way however, I hope this is interesting to you.
I'm not entirely sure of that. Certainly big things would, but if you drop the domain initiate as the first level and replace it with another type of thing, and then add a few more subtle differences, maybe in the saving throw progression.