NOTE: In my personal experience I've been able to do this trick much more consistently once I got a 75hz monitor with a 1ms response time (for reference my old monitor was 60hz with a 16ms response time)

Instead of waiting for the three pipe men to go into their pipes or grabbing the invincibility powerup, make sure to jump right before the little bush next to the first pipe, and immediately jump when you reach the very first few pixels of the second pipe man. Jump again once you land so you hit the ceiling of a floating platform, avoiding the Koopa. Make sure to do a very long jump when jumping across a gap so that you don't hit the next Koopa. Then do another very long jump to avoid the goomba before the staircase.