Chosen Solution
My head phones only work with the plug just slightly pushed in, I replaced the headphone/charging cable with a brand new one. I am still having the same problem. I tried 3 different headphones, one being the Apple supplied ones with the same results. With the headphones plug pushed in all the way like all I hear is static. Since it is a new jack with the same problem what else could the issue be?
Has your phone been repaired recently? There are some tiny passive components near the battery connector that are in the microphone/sound circuitry. If one or more of these has been pried off by accident, it could affect your headphone jack. Inspect visually or with magnification. Also, small parts are notorious for poor quality. Did you order them from a reputable supplier? It’s not at all unlikely that your replacement headphone flex is defective.