How to adjust the fill

[PixLab]

I would have put this "fill" on a layer under
Having said that, and because the font you're using allows for some trickeries  

If it's not on a transparent layer
Select the white edge (fuzzy tool)
then Select > Remove Holes
Then Select > Shrink ( few pixels up to the green thing)

You have selected your green thing  

If it's on a transparent layer
Alpha to Selection
Select > Shrink (then it's like above)

EDIT: Ok it's transparent
2 step as letters don't have same size

1)
Alpha to selection > Deselect the 2 big letters > then Select > Shrink
Do what you need to do
then 2) for the big letters
again > Alpha to selection > Deselect all the little letter > Then Select > Shrink
Do what you need to do

I did try on big letter and shrink by 13 pixels seems to be good