I think there are dog trackers on the market that use GPS + Cell. At least there are motorcycle trackers.
Generally the problem with WiFi is if it's close enough to work the dog can usually hear you. You could use a directional antenna but an extra 10db isn't going to buy you much. And at 2.4GHZ WiFi gets attenuated rapidly when not line of sight. Same is true for anything else in the 900MHz/ 2.4GHZ bands. Cellular gets around this by putting their antennas up high to clear local clutter (buildings, trees, hills, etc).
Problem with Cellular is the power drain and the monthly bill. On the other hand, probably will work. Cellular coverage is pretty good everywhere now. If you want to mess about I think there are modules with Cellular and GPS.
You also might be able to use a 900MHz solution[1] where you just send a ping of the GPS coordinates every so often. If you up the power to the max allowed (Like 20-30db) and drop the data rate down to a few hundred/thousand bits per second. You could probably go two to three miles. I think your current choices here are some sort of FSK radio or the newer LoRa radio. The reason is FSK radios and LoRa both allow you to turn the bandwidth way way down. The lower the bandwidth the better the signal to noise for the same transmit power.
[1] The lower the transmit frequency better it will propagate through clutter on the ground. But lower frequency means a bigger antenna.
PS: If you're messing with RF using module or manufactures dev board is the way to go. It's very hard to tune the RF path and antenna without experience and a spectrum analyzer.
PSS: I've found the transceivers from SemTech to be pretty bulletproof. They have their own power regulators on board which makes them insensitive to being powered by crappy supplies.