Selecting reamer size

Could someone explain me the process of selecting the reamer size for a barrel...my .223 shoot not so great n I imagine that the reamed opening is a few thousand big n the jump slightly deforms the bullet..the guy that did the barrel didn't use a reamer..the barrel is long so it could be cut again I'm planning in order different barrel later n I want to be able to purchase my reamers to ..any help is greatly appreciated..thanks.. 
 
For a pellet barrel, a standard reamer at 0.1 to 0.15mm over the nominal caliber works well. So for example in .22 cal (5.5mm), a 5.6mm is a good choice for reaming the leade. It will just wipe out the rifling. Then break the sharp leading edge of the rifling with a rubberizied polishing bit and/or wet/dry sandpaper so the pellet head can swage gently in the rifling rather than getting shaved.

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 your .223, 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. 
 
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This is what I'm using the 3 are 60g -- 3/4" long..there is a Ebay seller of reamers or I could order from Pacific tools..I don't mind price I need perfection..😁..could you post a pic of what I need to order..thanks a lot for responding..👊..I need to purchase them n take the tools n barrel to the machinists so he could doit again..is a airforce gun ..when you incert the bullet if you pin the bullet in the rear you could move a bit up n down..it should not suppose to happen..he machine me a pellet barrel n it works great .. but the bullets are a bit loose..
 
Well I'm not sure I can help you in the manner you seek. The key to success here lies not with the tooling. The tooling is the easy part. What you need is a competent machinist/gunsmith that knows how to approach it. It isn't hard to do but it's a one-off job that takes time, which is why I began describing the DIY approach. If that's not in the cards, I think you will be better served taking it to a good gunsmith.
 
Here in Puerto Rico I really don't know nobody in air gun Smithing just him the help the guys in FT .. with repairs n some custom jobs he is retired so he has all the time..but he is used to work in pellet guns..a bullet barrel is something different for him, your advice was great I could only need a pic of the reamer needed to se how it should look...in a airforce gun it should not be a problem since it has no special cuts needed or orings just a exact depth cut..since the barrel is .223 should I slug it sized .224 or .225...thanks 
 
Example #3 taper pin reamer:
https://www.ebay.com/itm/3-High-Speed-Steel-Straight-Flute-Taper-Pin-Reamer-NEW/324119549984

Again, the large end of this size reamer is 0.229" which is probably just a smidge bigger than you would want the leade to be for your .223 cal. So the machinist will need to first grind off a portion of the flutes to control the maximum diameter it will cut. If you were to do this modification in a home workshop, you might simply load the reamer in a drill press and use a Dremel tool with an emery wheel to grind it. A machinist will likely use a tool post grinder to do it.

Or you could use a metric one sized 5mm but there will be more material to remove, the large end being 0.250" (6.36mm).