If you get creative and mount the feed in such a way that you can slide it in towards the dish, you can select your beam width and effectively gain at will by adjusting the illumination of the dish.
Don't get too creative; the focus point exist only in one place. Gain will collapse rapidly and radiation pattern gets distorted and that makes the dish more prone to receive interfering signals around. This is of course highly depending on feed system itself.
(If you only
illuminate 4 feet of your 12 foot reflector, you should have the same gain and beamwidth as the four foot dish, just with a longer focal length)
Better use narrower beamwidth at the feed. (Underillumination) This will not degrade the performance of the dish. Only gain will be lower but beamwidth and even F/B-ratio can be increased. To illuminate just 4 feet area use narrower feed like helix. Calculate (or guestimate) the turns needed. Please read Paul's W1GHZ excellent "Online Microwave Antenna Book".
73s, Jason N1XBP
73s, Jari OH3UW