Why not use hash table with overflow area?

Question

Seems that using a hashtable with an overflow area (either within the table or another table) is not suggested as part of usual hash table implementations (linear/quadratic probing, chaining etc).
I was wondering why not?
If we dedicate a fixed size for overflow e.g. H then we have all the benefits of linear probing but without having to do O(N) searches where N is the size of the hash table but O(K) searches which is the size of the overflow area which could be kept small (in relation to N).
Is there any problem with this approach I am not seeing?