The fan-ownership campaign group announced on Tuesday afternoon it had purchased an extra 450,000 shares from Beaufort Nominees.
Easdale - whose brother James is among four directors who face being forced out of the Blue Room if Dave King's bid for power proves successful at the Ibrox vote on March 6 - owns 6.45 per cent of the Glasgow giant.
But the McGills Buses tycoon - who sits as chairman of Gers' football board - also held proxies for another 19.7 per cent, including those belonging to Beaufort.
King needs to win a majority vote if he is to replace current directors David Somers, James Easdale, Derek Llambias and Barry Leach with former Blue Knight Paul Murray and ex- Tennent's Brewery boss John Gilligan.
The RST hope their move will now speed up the South Africa-based millonaire's march to power.
Their latest purchase made through their Buy Rangers scheme means they now own 1,725,009 shares - or 2.12 per cent of the club.
They also hold another 1.7 per cent in proxies.
RST spokesman Chris Graham, said "This purchase makes us the largest fan group shareholder in the club and the 12th largest overall. We are also increasing our proxy block on a daily basis and will update this at the end of the week. We currently hold 3.8 per cent of the voting power for the upcoming EGM.
"We have shifted our focus in recent weeks to gathering proxies but also to targeting shareholders who we believe are aligned with the board but may be willing to sell. Purchasing shares off the market only takes you so far and we wanted to try to make inroads into the opposition's holding as well. This purchase is the culmination of two weeks of hard work on that front.
"We are delighted to have secured what we believe will be over a 1 per cent swing in favour of the nominee directors. None of this would have been possible without the fans who contribute to the RST and Buy Rangers. If they continue to join the RST in numbers and contribute to Buy Rangers then we will continue our efforts to reclaim our club."
Fellow fan ownership group Rangers First announced on Monday it has extended its own shareholding to 1.6million shares. It has also been given another 410,000 shares in proxies.