When you say you put a new recoil on do you mean a brand new one or a used one?
I ask because if it is used then the cord may not be long enough. If so then you will need a longer chord. I usually fit a 3ft long cord to a GX360 engine. Because i can get a decent pull to it. It should give you around 3 revolutions of the crank on a normal pull, which should be plenty to start the engine.
If it is a new recoil then it maybe the recoil spring inside has been allowed to unwind a turn or two. Or even been over tightened when the cord was installed.
You will need to change the cord. See parts list attached for each item.
Remove the recoil assembly (no/1) from the engine & cut the old cord off. Make sure the recoil spring (no/6) is fully unwound before threading the new cord in. You will need to remove the cover plate & bolt (no/11 & 8) on the recoil pulley (no/3) to be able to thread the cord in to the guide hole. Then knot the cord on the excess end that went through the plastic pulley. Making sure the knot is tight so not to pull back through.
Once you have done that, replace the cover plate & tighten the bolt gently (no/11 & 8). Then, turn the assembly over & feed the cord through the hole on the outer case (no/2) of the recoil assembly, do not pull it all of the way through though. Leave around 12in slack between the plastic inner pulley (no/3) & the guide hole on the inside of the outer recoil case (no/2). You need this length to be able to wind the recoil spring back up.
On the plastic inner pulley (no/3)you should see a small oval hole cut in to the outside edge. Turn the plastic pulley (no/3) counter clockwise until the oval hole is inline with the new cord & the guide hole of the outer casing (no/2). Then with the slack you have on the cord, pull the cord in to that oval hole. Now you should have a length of cord that you can use to wind the recoil spring (no/6) on with.
Wind the cord counter clockwise around 5 or 6 times, trying not to let the cord go unwinding the spring. Then feed the cord through the plastic handle (no/13 & 14) that you use to pull start the engine. Hold the cord you have pulled through the handle & release the plastic pulley on the inside of the recoil assembly. Gently allowing the spring to unwind.
With any luck, you should be able to pull the handle out far enough to give you around 2ft of pull & then allow the recoil to pull the cord back in until the handle is flush with the recoil housing. If this is all good, knot the end of the cord so it will not pull through the handle & push the cord back in, Then give the recoil a couple of hard pulls, making sure it rewinds all of the way back in. Then reinstall the recoil on to the engine. Hopefully you will now be able to give the engine a good pull to start it.
I hope this helps you & good luck