Line Up the Flyback
Before you even try to get the new flyback in place, make sure the pins are all straight. Out of the 4 flybacks I have replaced in the past week, all of them needed pins adjusted.
Once you get the new flyback in place, hold it firmly in place and "tack" solder two opposite pins. Once those tacks are in place, solder the rest of the pins and then be sure to finalize the tacks.