Any ideas on improving accuracy on a .22 cricket putting out 16fpe?

One thing that is cheap and is sometimes the problem is the hammer tension spring needs to be replaced. I would try that first. 

Actually, let’s ask a couple things. Is that the factory gun, with factory spring? If so, for sure change it.

If gun was shooting 18s at say 850+ before, there’s no reason for switching to lighter pellets just to achieve speed, unless you just want to use lighter pellets. 

If you are handy, you may want to tear it down, clean, replace all O rings, & hammer valve, rebuild and shoot again. 

Otherwise, may need to send to someone near you. I’m in GA so I buy and service my guns with Charlie at GA, but there are guys on here that can guide you along the DIY if you feel froggy. 
 
You cut the output in half, but used the same pellets? Yup, I am not surprised that you have issues. Try the cheap-o CPHP or CPDP next. At 6.00 for 500 and available at most Walmart and tractor supply, you can't go wrong.

Clean the barrel (my biggest error). Slower FPS means the pellet will be in the barrel for longer ttme, double down on your shot follow through. Hold the trigger against the back stop until you hit the target. It sounds funny, but it eally helps me hold position better.
 
So there is a bit of a question of whether the issue is from the air side of things or the pellet side of things. I'm essentially repeating what has been said above, but just to put it into context and give you a few concrete things to try:

Are you getting vertical stringing in groups which would suggest velocity inconsistency? Generally you can't turn down a hammer rate that dramatically and expect your valve opening to occur as properly and regularly. Hammer force is balanced against valve opening force which is based in large part on plenum pressure. You might want/need to turn down your reg a bit. 



But the other side of it is, from a dynamics perspective, pellets tend to want a certain forward velocity and rotational velocity in order to be aerodynamically stable. So JSB's 13 grain pellets as you suggested might be just the ticket. 



Quite frankly, my recommendation would be to sit down with front and rear bags and just shoot some groups. They might provide something diagnostic. 
 
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Moe,

I detune half the rifles I buy (for pure plinking purposes). So MY first question to you is how did you 'detune'? Just reduce hammer spring tension? So maybe 2 questions... have you shot it over a chrony to see is the shot string is still good?

If your change was the hammer spring, and you really want to end up with around 16fpe, thats more than I would ask from that alone. I would think you need to start by reducing the reg pressure. Not really hard to do, just takes some patience.