It is most common to do any% Yoshi's Island without 1-1 warps (termed "warpless"), as this is how the Japanese runners compete. The warpless strat for this level has several interesting tricks. First, the pipe glitch can still be used to warp to directly to the "Tetris" autoscroll room instead of jumping on the bouncy arrow over the save ring, since only 1-1 warps are banned. After loading in on top of the pipe, simply press down and jump on the same frame as illustrated below. This saves between 1-2 seconds, though will cost time if you accidentally re-enter the pipe.
After the autoscroll, the falling rock section can be skipped by bouncing off of any of the floating shyguys or jumping off any of the floating crates. It is recommended that you find a spot to jump such that you can just hold down forward and jump and the autoflutter will bounce off of one of the flying shyguys. A couple of strats are shown in the following GIFs:
Featured in the video below is the warped route for this level. It makes use of the pipe jump to activate the 1-1 warp glitch. Completing 1-1 saves about 45 seconds over completing 2-1 normally, due to the long tetris autoscroll.
Please see the section on warp glitches for information on activating the glitch, and surviving the subsequent fall into level 1-1.
The baby mario section requires using a visual or audio queue to jump at the correct time to be able to get over the ledge after transforming back from super baby mario. Alternatively, it is possible to perform a vertical gatehack as shown in the clip below:
Beware breaking your eggs on the stone blocks for the final star cloud.