What kind of carto? For plain 1ml stuffing-type cartos I don't need a syringe. I also don't think of it as refilling since if you run a carto dry you will singe it and ruin the flavor. I keep my cartos damp. This is how I fill, cleanup, settle, and keep damp.
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How I fill V4L/Volt/Bloog KR8 cartos and Boge 510 cartos
With the carto off the battery... Pop out the soft cap with an unbent paperclip in the mouth hole, very careful not to catch stuffing or the wire loop.
Shake and open the eliquid
Fill
Hold a bit of tissue on the threaded end hole (or use the carto condom spike to block the hole),tilt the carto, and add 18 drops per ml of capacity into the stuffing (Boges and most KR8 cartos are about 1ml). You may need to wait a bit for the first few to soak in, then continue. After getting the 18 in, wait 15 seconds or so then see how many more drops the carto stuffing will soak in easily.
Cleanup
Clean up the threaded end, tilt open end down into tissue, puff into threaded end to clear air channel (Boges and the KR8 cartos I use have a defined center air channel and I want to see light through that) While the carto is still open end down, touch a soft bunch of tissue to the top of the stuffing to get the loose juice - this will help prevent leaks.
Settle (first fill only)
Cap and let it settle - I lay it on different sides and on the ends for 15-20 minutes to make sure every bit of stuffing gets damp so I won't singe the carto. if you use thicker juice than me (I use 20% Vg or less) then you may need to wait much longer.
While vaping
While vaping - any time the vapor seems light, dry, or tastes the tiniest bit off or loses flavor, or if the carto is getting abnormally warm, I add liquid. I also occasionally look at the stuffing (especially before putting a fresh battery on or before going out) and if the top of the stuffing looks white, I add some juice. No settle needed when just keeping the carto damp. And if you do not keep it damp or if you use juice that is too thick (too much VG) you will singe the carto and that adds a bitter taste.
I use a simple plastic or delrin drip tip in place of the end cap so it is easy for me to remove that to look at the stuffing or add juice. I use thin clear juices that leave the stuffing white.