Moving for what was a sizable fee at the time of £1,000, he was converted from his original position of wing-half but it took a while for him to make his mark at the club.
With Gers beginning to dominate the Scottish game, they were winning titles galore but in his first three seasons he played just 22 times.
His fourth campaign brought a little more action but it was in 1931/32 that Simpson really became a regular in the team.
Now 24, he claimed the first major honour of his career that season as Kilmarnock were beaten 3-0 in a Scottish Cup final replay.
That was to be the first of nine trophies Simpson won as a player, with league wins coming in 1933, 1934, 1935, 1937 and 1939.
He was also present in three more Scottish Cup victories, the back-to-back triumphs of 1934, 1935 and 1936.
In total, Simpson played a total of 329 competitive games for Rangers – the majority as captain – and scored five goals in matches for the Ibrox team.
His son Ronnie also had a career in football and played as a goalkeeper for Queen’s Park, Third Lanark, Newcastle United, Hibernian and Celtic.
Jimmy Simpson (1927-40)
Appearances: 329. Goals: 5.
Honours: five league titles, four Scottish Cups.