You turned me on to something. I began Googling and found some interesting things. I hadn't been using my iPhone contacts except to look up old number "write them down" (duh) and enter them into the dialpad on Skype (I have SkypeOut).
Turns out you can make contacts in the iPhone use Skype: https://iphone-tricks.com/tutorial/5372-default-app-for-outgoing-calls
You can also sync Evernote contacts with iPhone: https://www.google.com/search?hl=en&ei=xrP1XJiOPOuQr7wPrZ6CiA0&q=sync+evernote+contacts+with+iphone&oq=sync+evernote+contacts+with+iphone&gs_l=psy-ab.3..0i22i30.58066.69415..69610...3.0..0.166.4192.13j24......0....1..gws-wiz.......0i71j35i39j0i67j0j0i20i263j0i8i13i30j33i22i29i30j0i13i30.hDLx56D39gU#kpvalbx=1
You can also sync contacts in multiple devices using iCloud: https://support.apple.com/en-us/HT205754
So the solution seems to be to use the iPhone contacts as the primary and set it up to dial out using Skype.