Somehow I missed that one. The early-2013 did not have 128 GB offered. Apologies.
128 GB is too small today. You don't need cleanup advice (though it would not hurt) - you mainly need a bigger drive.
The MacBook Pro uses a proprietary SSD controller - it will not recognize standard M.2 sticks. You need a compatible proprietary stick.
Bizon offers an Apple original 512 GB for US$400
OWC offers compatible sticks in different sizes
up to 2 TB. Their 480 GB starts at US$259.
Transcend offers compatible sticks in various sizes
up to 960 GB. Their 480 GB runs US$400.
Any of the above part numbers can be used to search eBay. Someone local may offer a better price.
Sintech offers adapters that allow some standard M.2 sticks to be installed. adapter 1
US$14, adapter 2
with backplate US$19. Check the compatible M.2 sticks listed for each adapter, they appear to differ.
Be sure to make a USB OS install drive before starting the upgrade. Along with a Time Machine backup. All the above options come without an OS pre-installed. Some options include a USB case for your old stick, which would be bootable.