That's up to you. I have a cheap hybrid, about a 22 mile route, and no particular desire to switch. I'd move to skinnier tires, but if the bike fits I'd just keep riding it.

I don't really believe that a different bike would make me enjoy the ride very much more, and I have other things to buy.

There are other people who seem to buy a different bike for every possible road and/or weather condition, and who am I to say that's wrong, it's their money.