Unique hashes are not something that we are likely to run out of in the near-to-medium term future, unless huge advancements in computation are made (in which case we'd have other issues to worry about).
When Bitcoin adds a zero to the front of the target, it is not a zero in hexadecimal (how hashes are usually visualized), it is adding it in binary. The difference is huge:
0FFF -> 00FF
means this in binary:
0000111111111111 -> 0000000011111111
What really happens is this:
0000111111111111 -> 0000011111111111
Or in hex:
0FFF -> 07FF
This means it will take 4 times as long to reach your point of concern as you thought. Also, let's look at the block hash you posted:
There are currently 18 leading hex zeros, which translates to 72 leading binary zeros. This leaves 184 bits (256 - 72) for a unique hash value, or 2^184 unique hashes. So right now, if the difficulty never increased and assuming a 10 minute block time, that still leaves us with 4.66*10^50 years worth of hashes (which is roughly 5 time the age of the universe). When the difficulty doubles, we now only have 183 bits of hash space. This means it's about 4.66*10^25 years, or about 2.5 times the age of the universe of hash space left. As you can see, it will take some time for the difficulty to reach a point where hash space is a reasonable issue.
However, the real problem is not with total hash space, but with probable collisions. Let's say that we have gotten to the point where there are only 56 bits available for hash space. First, this would take a truly massive hashing increase in the Bitcoin network to ever come about (difficulty would have to double 128 times), but we'll ignore that for this example. While 2^56 may seem like a small hash in modern cryptography, for this use case, it's not. We still have enough hashes left to cover the entire age of the universe, but again, we are concerned with someone finding a block with a hash that matches an existing block (or competing block).
So what would we do? I'm sure there'd be some debate if this ever came up, but I think the simple solution would be to reject that hash, just as if it were above the target difficulty. Miners would be forced to continue hashing until another block was found. After all, there is still plenty of room in the hash space, even when reduced to 56 bits (which is currently preposterous). Such a change to the protocol would only be a soft fork.
To summarize, even with our best estimates of future computing technology the remaining time for Bitcoin hashes is best measured in multiples of the age of the universe. Even if this became an issue, a simple soft fork could fix it. There's really not much to worry about here.