After previous problems with compression I got my FE350D rebuilt by professional shop. Now I put a brand new carb in it. Replaced air and fuel filter. Replaced fuel pump and all the fuel lines. New spark plug and new ignition coil. Now to start it I have to press on the accelerator a little and keep it presses for the engine to run. As soon as I release the accelerator it dies. What else should I check for? Just in case I performed leak down test and it's good. 

Sounds like a fuel issue, lack of...
 
Did you try turning up the idle screw (basic but have to ask)?
 
#92064 is the pilot jet: 
https://www.jackssmallengines.com/jacks-parts-lookup/manufacturer/kawasaki/4-stroke-engine/fe350d/fe350d-as07-4-stroke-engine-fe350d/carburetor 
 
What are the leak down #'s? 
Did you have this issue before the rebuild, ?
 
Sometimes, the Chinese knock off carburetors are not so good and can cause issues by itself..
 
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Anything I replaced in that gator always purchased from JD. I didn't touch the idle screw because service manual says not to.