The correct solution to the interpolation problem is a class of specially designed FIR filters called Fractional Delay Filters.

Each Fractional Delay Filter can calculate the samples at one set point between any two samples. For instance, you need one filter to calcaulate a half sample delay, another to calculate a one third sample delay and so on. Theorectially, to be able to calculate a sample at an arbitrary fractional delay you need an infinite number of FIR filters.

The short cut is to use use a small set of Fractional Delay Filters and then use linear or polynomial interpolation between the point provided by the fractional delay filters.

For instance if you have Fractional Delay Filters which can calculate samples delayed by 1/4, 1/2 and 3/4 of a sample and you need to generate a sample delayed by 1/3 of a sample, you need to calculate the samples delayed by 1/4 and 1/2 of a sample and then interpolate between those two.