The Problem

To get more dirt in vanilla there are two ways (other than mining):

1. Combining 2 dirt with 2 gravel to make coarse dirt and then hoeing the coarse dirt to turn it to regular dirt.
2. Growing Azalea trees on moss to get rooted dirt and hoeing that.

You can't use method 1 because there's no gravel here and also there's only one dirt block on the entire island. Method 2 is not possible either without moss, which you can get from wandering traders, but that's a slow and RNG dependent process.

You know what isn't slow and RNG dependent?

Anvils

You know and love them. If you've been reading the guide in order then you wouldn't know that anvils can also be used to crush and compress blocks. This is how we will obtain another piece of dirt.

Crushing

Crushing blocks is easy. Place a block of cobblestone. Drop an anvil on it. Now you have gravel. Get 4 blocks of gravel. Next crush one block of gravel into sand.

Compressing

With gravel and sand there's certainly much more you can do, but we want dirt. This is where compression comes into play. Place a gravel block down and place a sand block on top of it. Drop an anvil onto the sand. Congrats! You made a coarse dirt block! You should know where to go from here!