If you feed 'high energy' type feedstuffs then you have to continue to do so, i.e. once their metabolic rate has been increased from winter slow-down; if not, it is actually counterproductive, in that they will often move away to find a source of 'high input' foodstuffs.
It is seldom indeed that you will kill a deer with an empty stomach, usually a sign of something wrong with it; much deer range is to some degree deficient in one or more trace elements (-do the local farmers put out buckets of mineral supplements for their stock? If yes, you can bet it is not for charity, but good reason); most deer will of course happily use the available forage, but will get habituated to supplementing this forage with any source of good trace elements they are otherwise lacking, to help them fully convert the forage. This is the more cost effective strategy, but lower energy inputs will still maintain the deer, at a more natural (seasonally lower) metabolic rate.
You can spoil them with loads of food if money is no object, or you can supplement their trace elements to maximise their use of what is already available. Deer feeding on 'new' feedstuffs can literally die of starvation even with a full stomach though, owing to their gut flora being unable to convert to deal with the new foodstuff.
I brought a half tonne of mineral blocks to one head stalker I've supplied the past twenty-five plus years on an estate last weekend, he told me this winter his deer are eating much more silage than in other years, which he put down to the very wet autumn and early winter, which he considers basically washed much of the goodness out of the forage; it's conjecture, but all the same makes for a reasonable stab at the reason for their increased appetite for the silage.