Type of Coverage
The type of warranty you will probably be looking to extend is the basic warranty Powertrain warranties generally last a lot longer and include the least likely to break down mechanical parts of the car. On the flip side, they are also the most expensive to repair if they do breakdown.
A Keeper, or Sell
Of course, this depends on your needs. If you do not plan on owning the car for that much longer, it probably is not worthwhile to purchase warranty coverage for a relatively short duration. If the car is a keeper, then purchasing an extended warranty that will cover the next 100,000 miles is probably a good idea.
The model of the car should also be factored in. A reliable model is less likely to have "normal" mechanical failures. If you plan on selling the car shortly after the basic warranty expires anyways, and the car model is known for being solid, there probably is not any reason for you to spend money on extending warranty coverage.
Let's say you originally decided that you wanted to keep the car and extended the warranty. Eventually, because Santa Claus was generous, the decision to sell the car anyways is made. Well, you already purchased the extended warranty, but whether or not the extended warranty is transferable may influence your bargaining power at the table. Extended warranties that are transferable when the car changes ownership are preferable; asides from being a selling point, having a still valid warranty on a car you give away as a gift can be highly convenient.
As always, read the fine print on the contract. "Per visit" or "per repair" deductibles; "mechanical breakdowns" or "wear-and-tear breakdowns"; "long block" or "short block" (for the powertrain)--all of these things have different meanings in the world of warranties. A "per visit" deductible only runs you one deductible every time you visit a shop; "per repair" will run you a deductible for every single part replaced. "Mechanical breakdowns" may be used as a loophole for warranty companies to claim that only major, completely out of the blue, mechanical failures are covered; "wear-and-tear" refers to any mechanical failure that is a direct result of normal wear-and-tear.
What is Covered, Where it is Covered, How it is Covered
Know precisely what is covered, and then have them spell out, specifically on the contract, what is not covered. Also be sure to retain the flexibility of sending the car to any dealership or mechanic of your choice, or at least a very wide range of choices. Some warranty contracts only offer reimbursement rather than direct payments; direct payment is always preferable because reimbursement can take months, and may require a substantial amount of cash upfront.