It depends on which country.

If you overstay a US visa, awaiting a new and better visa, and you get caught you can be deported to your country of origin, and banned from reentering the US for a number of years.

If you discuss this with folks at your country's embassy or consulate, you may be able to get a short extension of your visa, enough to allow you to leave legally, as opposed to being deported.

Dave