Please provide more details, what phone do you have?
Is your phone an older model with a microUSB port in it or is a newer model that actually does have a USB Type-C port? If the former, no just using a Type-C cable isn't going to make your phone charge any faster -- all things being equal (wire gauge) the difference between a microUSB cable and USB Type-C cable is one has a microUSB plug on one end and the other has a USB Type-C plug on one end. Granted, USB Type-C charging will involve higher levels of current so it is faster but that's an aspect that's in the phone, not the cable.
Not knowing what kind of power adapter you typically plug your phone into nor the head unit specs in your vehicle, the odds are your home power adapter is going to provide more current than a car's USB port, so presumably that's also a no, it won't charge faster in your car.
Basically, stick with a USB cable that's directly compatible with your phone instead of adding in a mix-and-match, jury-rigged set up. Focus more on what kind of power adapter you have. If it's an older one that's just rated at 1 amp/5 volts by a newer one that's at least 2 amps/5 volts.
Also, since you didn't want to reveal what phone you have, that could be part of the problem too. All batteries age and degrade so that's a given. A dated battery won't be able to last as long as it used to, and it will take longer to re-charge itself.