Green Hill Zone
|Big Pink||Adjacent rooms||Morph Ball Room
After the room loads, do a full jump. The very first frame will be too early, but elsewhere on the ledge will work. This will allow for a mockball on the floor below the first pipe.
If you miss the jump, angle up, and fire a shot right as you reach the end of the second ledge in order to kill the Geega before you run into it. Then you can backup with a small jump into a bounceball.
Either way, you should be able to bounce under the first sidehopper, then stand up, run right, and start charging a shot. When you run off the edge, aim down to compress your hitbox to avoid running into the ceiling, then let go of down, and let go of shot to kill the second sidehopper. Start charging a shot again, and fire it when you're level with the top of the pillar to open the blue gate.
Jump over the pillar, charge another shot, and kill the grounded sidehopper. Run right, fire a shot, and start charging another. The last shot will hopefully open the door, so you can jump through the transition with a shot fully charged.
However, Space Jumping up the stairs with Screw Attack is only slightly slower. Attempting Reverse Slinky and failing it partway through can cost significant time, because you can't build enough run speed to quickly Space Jump up the rest of the way.