Are there gaps in what our ears can hear?
Short answer
No, there are no between-hair cell induced tonotopical gaps in frequency perception.  

Background
Young people are able to hear over a frequency range of about 10 octaves, with a frequency resolution of about 0.3% of an octave. So we would have about 2300 resolvable frequencies. This number is dependent on the sound level, as excitation spreads at higher sound levels, but it gives a reasonable starting point to answer your question.

The human cochlea contains on the order of 3,500 inner hair cells. Hence, according to this (highly simplified) calculation, it seems that the number of hair cells is redundant.

In other words, the notion of in-between hair-cell stimulation is stretching the limits of physiological frequency resolution a tad too far.

Reference
- Elliott &amp; Shera, Smart Mater Struct (2012); 21(6):064001

Source
- Cochlea.eu. Cochlea: function