Based on those relations there's nothing more that you can conclude. Consider the two extremes.
At one extreme $\pi_1$ and $\pi_2$ project onto the same subspace, in which case:
$$\langle {\phi} | \pi_1 \pi_2 |{\phi}\rangle = \langle {\phi} | \pi_1 |{\phi}\rangle = \langle {\phi} | \pi_2 |{\phi}\rangle \ge e, \;\; \pi_1=\pi_2=\pi_1 \pi_2.$$
At the other extreme $\pi_1$ and $\pi_2$ project onto perpendicular subspaces, in which case
$$\langle {\phi} | \pi_1 \pi_2 |{\phi}\rangle = 0, \;\; \pi_1 \pi_2 = 0.$$
Based on the stated relationships, the projectors could exist anywhere between and including these two extremes. If $e>0$, you could at least say that $\vert \phi \rangle$ is not perpendicular to either the $\pi_1$ or $\pi_2$ subspaces, but that alone still doesn't tell you any more about the value of $\langle {\phi} | \pi_1 \pi_2 |{\phi}\rangle$.
EDIT: DaftWullie's answer makes an important point that I missed. If $e>\frac{1}{\sqrt{2}}$ (with $\vert \phi \rangle$ normalized), $\pi_1$ and $\pi_2$ cannot be orthogonal projectors, in which case $e$ imposes the lower bound $2e^2-1$.