It is tough to get any price break in June. Everyone, especially the once tied to school schedules,

There is no formula whatsoever. Sign up for notices from the airlines you're interested in flying with, as well as Kayak and related sites, and just wait until something comes along that you can live with.
1) You should know that June is summer and fares will be their highest, with last minute discounts generally unlikely. Also, first half of June might be lower than later in June due to school's getting out about mid June.
2) If you do a search on www.kayak.com for various high and low season virtual home tours rates (compare March with June, for example)you can formulate an opinion about what fare is acceptable to you and then when you can check back frequently or when an alert comes and you see the fare is OK, go ahead and purchase virtual home tours then.
3) Think through your itinerary enough to consider a possible "open jaw" ticket. For example, if you intend to visit both Rome and Venice, then try pricing virtual home tours Atlanta to Roma with your return Venice to Atlanta. An open jaw ticket is not necessarily more expensive than a round trip ticket and can save you one night's lodging plus train and taxis that might make a material difference in both time and cost.
I bought my April/May 2013 tickets in June 2012. I wanted seats near the front. I don't look to see if prices have changed...I don't want to know and suspect it won't make much difference anyway. virtual home tours I like to have the tickets before I book hotels/apartments.
The minimum you should be doing is right now is to collect empirical data for your dates by signing up for a fare alert at www.kayak.com . You get email updates and a graph of how the fare is trending. I routinely have 20+ fare alerts active to look at the itinerary from many different angles using many different airports and different dates. When it comes time to purchase tickets, you have actual price trends to see if you are trying to buy them at the high or low end of the actual price trends (not last year data.) I have the alert set to notify me daily.
There are two ends that are "usually" high. Very early and very late. Early people are those must MUST travel on these dates (price virtual home tours inelastic) and airlines know there is no need to offer incentives. Late people those who HAVE to get to the destination ASAP (price inelastic). Again, no need to offer any price breaks. In between, which are many months, the airline "might" offer competitive prices to entice people who would not have chosen virtual home tours to travel or that airline at that time to that destination.
It is tough to get any price break in June. Everyone, especially the once tied to school schedules, travel in June and the airlines make sure to capture the premium revenue streams. Looking at the prices virtual home tours for mid May, mid June, and late July would show you just how much premium you will get hit by traveling in June.
Once you've identified the specific flights you might be interested in, go to the carrier's own website and start the process you would go through to actually buy the tickets. At some point short of pulling virtual home tours the payment trigger, you might be able to see how many seats are still available on that flight. If hardly any seats have been purchased, there's no reason to buy the tickets at that time. If they're starting to fill up and you really want that flight, go ahead and buy.