  The residual interaction between composite fermions (CFs) can express itself
through higher order fractional Hall effect. With the help of diagonalization
in a truncated composite fermion basis of low-energy many-body states, we
predict that quantum Hall effect with partial spin polarization is possible at
several fractions between $\nu=1/3$ and $\nu=2/5$. The estimated excitation
gaps are approximately two orders of magnitude smaller than the gap at
$\nu=1/3$, confirming that the inter-CF interaction is extremely weak in higher
CF levels.
