Prior to this year, when June 1st arrived teams could cut veterans and the remaining portion of their signing bonus would be divided into this year and next instead of just this year only (if cut before June 1st). So if you had 4 million remaining from their signing bonus... you would have 2 million hit this season and 2 million the next if cut after June 1st. However, if cut before June 1st, all 4 million would be accelerated into this year's cap.
Teams were holding on to players (until after June 1st) that they had no intention of keeping, simply in order to divide their cap hit into two seasons instead of just one. So, the NFL, in order to give those players more time to catch on with other teams... before all the cap dollars are spent and before teams have pretty much finalized their rosters... said that two players could be designated as post June 1st cuts, even if cut prior to June 1st. That allows teams to divide the remaining cap hit into two seasons while still letting the players get on the market with plenty of time to spare.
That is the sum of the change... at least that's how I understand it.
So, I would say that most teams have probably cut who they're going to cut and the chances of landing a solid player after June 1st would be slim. I'm not saying that it can't happen... but most teams likely played under the spirit of the new rule and cut those guys loose way before now. JMO, of course.