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Lead in taper

For a slug barrel, a slight taper of about 1° is preferred. Custom barrel reamers are pricey but a tapered pin reamer (1.2° included angle) is a good and inexpensive way to do it. Figure out where on the taper is the maximum diameter you need and grind away the portion behind it.

For example, for a .22 cal barrel, a #3 pin reamer would be good. It tapers out to a max of 0.229" so you wouldn't have to take off much. For a one-off barrel, run it in somewhat short of what you think you'll need and try seating a slug. Sneak up on the depth you need, such that your bolt probe seats it just into the taper. 

Refer also to this thread for more DIY discussion.
https://www.airgunnation.com/topic/selecting-reamer-size/
 
Just contact AE Swede and ask about sending your barrel to him. He has made several slug gauntlets and knows how to do it. He doesnt offer his service on his website, just contact him and ask if he still does it. Super nice guy and very trustworthy. Priority mail, well packed in a cardboard or PVC tube, you wont have any issues in shipping. Its going to be cheaper to send it off than it will be to buy the tools to do it yourself and possibly mess it up. 

He made a long range slug gauntlet, I want to say he was shooting 38gr NSA and taking prairie dogs to 250-300yds. 

If this is a .22, mine shoots 17.5gr NSA just fine without barrel mods. 0.6" group at 75yds last time I grouped it.. it doesnt have the power for the heavier slugs IMO.. I tried 21gr and 23gr, along with 25gr knockouts and they all destabilized at 75 and were going all over the place. 1400psi reg, hill hammer kit shimmed up 3 shims and I am getting 815fps with the 17.5gr.