The 185/65R14 will be about 3/4" taller than the 175/70R13. The 3mm is in the tire height, but the 1" difference in the rim makes it a different story.
You mentioned that you still need to check the rim you now have as the spare.
Did this car have a 14" option and therefor does that rim fit the hub ? Maybe it is a fit for the car and all you need is a suitable tire.
Watch out for those used tires though. Age alone is quite a monster under the bed. A tire with few miles might look ok but its age will quickly rear its ugly head when you start that dried out old thing rolling and flexing after all this time.
I have a 1979 that has merely 71,000 total miles on it.
It is on its THIRD set of tires.
— AGE ! —
I’m going to make another suggestion to you here. Look at the pick and pull for a newer Elantra with either 14 or 15" wheels. A good quality 13" tire is getting hard to find as most are being discontinued. Even good 14" tires are getting hard to find, but there is some selection in the 185/65R14 size, but it is limited. 15" rims will open up the door to greater selection of premium or performance tires.
You may look at other manufacturers for the wheels as well, but make sure that it a FWD car and the bolt pattern and the hub size matches. Too many people overlook the hub size and end up with a bad ride because the wheels won’t center properly without hub rings.