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http://darksouls2.wikidot.com/online-matchmaking
^ has the summoning mechanics, exactly how they work, with and without the ring. It does become more permissive as you reach higher SM tiers, which have larger ranges. And the ring makes it more permissive still. But NG+ has nothing to do with it. Matching a high-SM character with a new one is never going to work, sorry.
http://steamcommunity.com/sharedfiles/filedetails/?id=419088382&searchtext=achieve
Scroll down to the multiplayer and co-op section, it shows the soul memory tiers and how far each multiplayer item reaches.
Let's say you want to invade at soul memory 230,000, that would be SM tier 13, 210-240k is your tier, cracked red eye orb let's you invade people in your tier to four tiers higher, meaning you can invade people from 210-400k, the engraved ring does not benefit the orb though.
Now suppose you wanted to be summoned or summon someone for co-op, let's say the host's SM tier is 3 and yours is 10, the small white soapstone only reaches 4 tiers down and 2 tiers up, but with the engraved ring, it reaches a nice, meaty 7 tiers down and 5 tiers up!
You can now connect until you level out of the host's range.
Sorry if this explanation was long-winded, I tend to ramble and lose my train of thought.
Lastly, the engraved ring in no way negates the soul memory tier, that would be extremely overpowered.
Engraved ring: Increase your soul memory tier range.
Agape ring: Don't gain any souls, and don't gain any soul memory to freeze you at your current tier.