That spur was actually the original Hempstead branch. Trains left the main line at Mineola and headed south to Hempstead. At that time, A.T. Stewart's Central Railroad of Long Island parallelled the LIRR main line (It forms today's Hempstead branch between Floral Park and Garden as well as the rest of the so-called Central Extension, and the single track Central Branch between Bethpage and Babylon). The Stewart railroad had its own Hempstead branch and Hempstead station, slightly east of the LIRR's.
When the LIRR lines were consolidated, the LIRR chose to connect with Hempstead from the former Stewart line. But the LIRR branch followed the alignment of the LIRR's own Hempstead branch to a point just east (geographically south) of Country Life Press station.
Today, the S-curve between Country Life Press and Hempstead marks the point at which the LIRR moved over to the former Stewart right of way. The LIRR even used Stewart's original Hempstead station for a time, which was actually south of today's station, on the site of today''s Hempstead Bus Terminal.
I rode that line in September 1959 on a fantrip using the railroad's two Budd RDCs. The train ran Jamaica to Greenport, then back to Mineola, crossed over to the eastbound main, then up the Oyster Bay branch to Oyster Bay, then back to Mineola, then crossed over again to the eastbound main, then down the spur to the former Hempstead Crossing. Only the northeast segment of the former wye was still there, so it put us on the Central Extension facing east. The engineer changed ends and it was back to Jamaica using the Hempstead branch to Floral Park, then the main line. Floral park was at ground level then. The beginning of construction of the grade crossing elimination project was around two years in the future. Bellerose station was a main line station, with a platform for the westbound local track, and a platfrom for the eastbound local track. I was all of 14 years old then.
Fairbanks-Morse forever!