I have a round-trip coming up and on the return flight I couldn't get a direct connection so I had to book a layover. I later decided I would like to spend some time in the connecting city, perhaps a few days. Is there anything wrong with me missing my connection flight (not checking anything) and fly home on a new one-way ticket a few days later?

I would need to buy the new one-way ticket in advance, but probably from a different carrier, so it wouldn't be apparent to the airline. If someone has access to info from different airlines it might be clear what I did.

I'm not sure I see anything wrong with what I'm doing, but is it possible the airline could charge me a fee for not taking my final flight? This doesn't make sense to me because I already paid for the ticket.

Thanks!