Enumerating set combinations in an order that maximises the number of previously unseen subsets
There are $\sum_{j=1}^4 \binom{11}{j} = 561$ smaller subsets, and each $x^\phi$ contains $\sum_{j=1}^4 \binom{5}{j} = 30$ of them.

If you put all $462$ $5$-element sets in a priority queue with priority corresponding to the number of subsets which haven't yet appeared, after each pop you have to check $30$ subsets to see whether they're appearing for the first time, and for each $k$-element subset that is appearing for the first time you have to update the priorities of $\binom{11}{5-k}$ sets. There's an easy upper bound on the number of updates of $25410$.

As a follow-up optimisation, once every smaller subset has been seen (which happens after you've removed the first 90 elements from the priority queue), you can just iterate through the rest. If your priority queue is e.g. a binary heap, this will save you a lot of $O(\lg n)$ pops.