Aside from a manufacturer's warranty, used car dealerships – in particular car supermarkets and independent dealers – also offer warranties either as part of a car's price, or for an additional fee. Often, they'll sell you a car with a short warranty on it (around three months) and ask if you want to pay to extend or upgrade the cover.
Alternatively, you can choose an after-market warranty which can be tailored to suit your personal requirements. However, your options will depend on the type of car you drive, its age, and your annual mileage. If you opt for this type of warranty and your car's older than three years, you'll be asked to provide your MOT certificate and prove that your car has been serviced within the past year.
If you've already got a warranty but it's up for renewal, it could be cheaper to purchase new cover as opposed to extending your existing one, so spend some time researching for the best deals.