How does hash table search scales with number of keys? Say, I have one of the standard implementations of a hash table: chaining or open addressing, and my hash key is a pair of strings. Instead of using the hash table with the pair as a key, I can use the first key to get to a "sub" - hash table, and then use the second key to find the element from it. 

How do these strategies compare, e.g. if one assumes the key pairs are more or less uniform?
In principle, they're similar in running time and space complexity.

In practice, the first one might have some modest advantages: possibly half as many accesses, less space overhead if some of the "secondary hashtables" were small (e.g., for a particular value of the first key, there aren't many possible second keys).

I would not expect to see a tremendous difference between the two, nor a difference in asymptotic worst-case running time.  Both have $O(1)$ expected running time for all major hash table operations, under some reasonable assumptions.