You've pretty much got two options the way I see it.
Option 1, you get creative to get the big card to fit in your small case. I've done the same thing many years ago, where I had a Dell slim case that sat horizontally and a video card designed for normal vertically designed cases. I had to remove the faceplate on the video card (the shiny silver part where the DVI/HDMI ports are. It is held in place by the little bolts the thumbscrews screw into) and leave the cover off the case. It just barely fit, but I was able to get it to work. This is not ideal, as you expose your computer to potential foreign objects as well as increased EMI, but it will work and is cost effective.
Option 2, buy a new case when you buy a new video card (most likely you'll need a new power supply as well, since it's doubtful the slimline power supply will be enough to power the new video card). Your Dell system should be using a standard ATX motherboard, which you would need to verify ahead of time. But if it is, then you will be able to move the motherboard and other components from the Dell case to your new case. Then, a standard video card like the GTX 660Ti you mentioned would work just fine. I personally did this same thing a year ago in order to put together a computer for my sister's children. This is the ideal solution, but it will cost more money.
You cannot fit a full-size card in a slim case. The only solution would be to replace the case. You cannot make the case longer or wider then it already is. The physical dimensions for the card are provided in most cases. You should make the required measurements to determine if its actually even a problem. – Ramhound – 2014-02-08T21:21:38.673