A network reset by you the phone owner may cure it, BUT if you can call your cellular provider on your phone do a reset, that works best. You call them. They do a reset and you shut off your phone for a couple of minutes to let the reset work through the network. Then power up the phone. That should fix the going right to VM.
To reset yourself:
Settings/General/<scroll down far>/Transfer or Reset iPhone/Reset <Not Erase>/Reset Network Settings<Not all the other stuff>
Screen goes black---WAIT a minute or so ---- the phone is "finding" the network.
Phone will power back up itself.
This also helps the phone FIND TOWERS in new locations; so if you travel do a Network Reset is each location you will spend time in, THEN reset when home.
Also, other members above had made good suggestions with a protocol (series of steps) to check various settings on your phone; the silence switch (top left of phone- orange bar shows silent--will go to VM), call forwarding, call blocking and so on. Run those settings down as well if a network reset does not work.