I also remember a rail spur which ran between Mineola and the cross junction north of Country Life Press. It ran south from Mineola, ran parallel to Franklin Avenue then went east towards the old Clinton Avenue station, which is now a Garden City fire house. I don't recall any passenger trains running on it. It did, however have a third rail.
An old map of mine shows the Garden City junction as a grade crossing with connecting spurs in all four directions. In other words, trains could have gone from Mineola to Country Life Press if this were the way the tracks were. This must have been before my time - I only remember three connectors at most - Mineola-Clinton; Garden City-Country Life Press (still in use); and Country Life Press-Clinton (I think the Clinton shuttle train used this). I don't remember a Garden City-Mineola connector. This same map also shows an eastbound connection between the Far Rockaway branch and the Babylon Branch at the Valley junction. The curvature of the current elevated structure tends to support the former existence of this.
Wayne