  Motivated by recent works on atom-cavity realizations of fast scramblers, and
on Cooper pairing in non-Fermi liquids, we study a family of solvable variants
of the ($q=4$) Sachdev-Ye-Kitaev model in which the rank and eigenvalue
distribution of the coupling matrix $J_{ij,kl}$ are tuneable. When the rank is
proportional to the number of fermions, the low temperature behavior is
sensitive to the eigenvalue distribution. We obtain a complete classification
of the possible non-Fermi liquid quantum phases. These include two previously
studied phases whose fermion scaling dimension depends continuously on the
rank; we show that they are maximally chaotic, but necessitate {an extensively
degenerate or negative semidefinite coupling matrix}. More generic
distributions give rise to "almost Fermi liquids" with a scaling dimension
$\Delta = 1/2$, but which differ from a genuine Fermi-liquid in quasi-particle
decay rate, quantum Lyapunov exponent and/or specific heat.
