You'll want a bit of a different sort of smarts for this
While controlling outdoor outlets isn't the forte of most smart-switch systems, especially those that rely solely on wireless technology, your "typical" options aren't the only game in town for smart-switches. In particular, the old power-line communication standby of X10 hasn't disappeared altogether; instead, it was succeeded by an updated protocol called Insteon, and that technology provides us with several advantages here, including remote modules that can go in fixture boxes and the ability to use reliable power-line comms to go places wireless systems just don't go.
In particular, I'd replace the existing switch with an Insteon switch, but leave the LOAD terminal on the switch unconnected, only connecting Neutral and Line. This allows the switch to be used as a remote, controlling an Insteon Micro On/Off module located at the outdoor box, which gives you your switched hot for the outlet. (Note that this circuit will need GFCI protection somewhere in line to protect the new outdoor outlet, by the way; if you have room and all the necessary wires at the wallbox, putting a deadfront GFCI alongside the Insteon switch there and feeding always-hot and neutral through it on their way to the fixture box is your best plan.)