Unfortunately You already have a fairly good processor and the i7s will probably still be pricey even used.
Probably better to look for a new motherboard and Coffeelake processor when the H370s come out. AMD may be a good option too.
Looks like you already have the bios needed for 4th gen K processors.
http://www.gigabyte.us/Motherboard/GA-H81M-D2V-rev-10#support-dl
Is there a lot that your current processor won't do that you want it to ? I note that it is still in the top rank in the Toms CPU hierarchy. Which means that upgrade even to the best newest processor may not improve your gaming performance that much.
http://www.tomshardware.com/reviews/cpu-hierarchy,4312.html