By the way, while the Michael's answer is absolutely correct and practical (you need a new lens!) it might be interesting to learn that you actually CAN achieve the same FF look without changing lens under some circumstances (other than stated in the question - it will not work with 600D and with street photography).
And yes, technically this is a comment, not an answer, but it would be too long ;-)
First, it might be possible to use a focal reducer.
For example I am using a Sony APS-C + Lens Turbo adapter that makes the 50mm f/1.4 lens behave almost exactly like on a full frame camera (but this is only possible with mirrorless, because of their shorter flange distance).
Second, taking the APS-C photo, you already have a fragment of the full frame photo, just go on!
Remember that the FF lens still creates exactly the same image when used with APS-C camera... you merely can't see the whole thing because the sensor is too small. So why not move the sensor and catch the full picture step by step?
Traditionally, this is achieved by using a tilt/shift lens (again, just a theoretical possibility, as this is not the kind of lens you have), but today the same effect can be achieved much easier with any lens, just by rotating the camera and adjusting the perspective projection in post.
By stitching the images, you get the exact equivalent of the FF sensor, we can even say the image WAS captured by the full frame sensor, just not all parts at once. Obviously, this method (known as the Brenizer Method) is not practical in all situations.