Space efficient "industrial" unbalanced expanders
Eickmeyer and Grohe (2010) prove that your candidate construction can be made explicit: take $d$ somewhat linearly independent linear hash functions $h_1,\dots,h_d$ and connect left vertices $v$ with right vertices $h_1(v),\dots,h_d(v)$. Eickmeyer and Grohe show that this construction gives $(k,\epsilon)$-expanders with left degree $d=k(t-1)/(2\epsilon)$, whenever $t$ is an integer, the left vertex set has size $n=q^t$, the right vertex set has size $m=dq$, and $q>d$ is a prime power. The hash functions $h_1,\dots,h_d$ are chosen in such a way that any $t$ of them are linearly independent.