Explain what you mean by you are “re-doing you cabinets”. Are you doing a complete kitchen remodel or are you painting the cabinets and getting new countertops?

After looking at your kitchen layout I would be highly inclined to move the refrigerator to where the pantry is and filling adding in counter space where the fridge is removed. It seems like a better kitchen triangle to move the fridge to that wall and it is also hidden from the living area.

It looks (to me) like the sink island is too close to the wall of cabinets. If you want some suggestions, we would want to see a measured/labeled to-scale drawing of your whole space and the adjoining spaces. What is your budget and what are you thinking of doing?
What are your issues besides the shallow pantry?

Finstads Carpet One: I am getting new cabinets and counter top. At the same time, I was wondering whether I should rearrange some parts of the kitchen. There isn't much room to move anything but I feel the little counterspace on the left side of the fridge is wasted space.

I've also thought about moving the fridge to the current pantry space, but I'm worried about the wall and opening the fridge door.

Felizlady: The clearance between the sink island and the cabinets is I believe 3ft which is the minimum. Not much room to move the island further away though :( As I mentioned, mostly changing counter top and cabinets; also removing the microwave range hood! my biggest concern would be just functionality and final look of the kitchen