For 38 cents, I can't even dock a star for this... To make it fit an iPhone case, just trim off the lowest part of the plastic don't worry, there's metal under there. Took me a minute with a pocket knife, and now it fits perfectly in an iPhone 6 TPU case, also from Monoprice.
Neither this adaptor nor the similar 7161 would quite fit my Speck CandyShell case, which has about an 8mm hole. (Both adaptors did fit my iPad Mini with case 9967.) However, I got a knife and carefully carved away the circular part of the plastic (before the main hexagonal part). After that, it does fit my Speck case nicely. The metal part beneath the plastic is about 6mm diameter. It's a shame Monoprice doesn't offer a version using their Designed for Mobile plug, because I got one of those cables also and it fit my case just fine.