My approach is generally to have meat-only stockpiles and fat/tallow-only stockpiles right next to (or surrounding, and even checkerboard-style) the butchers. (I find that I prefer making larger stockpile areas with a no-barrel policy is visually easier to police than the correspondingly smaller ones possible with the increased capacity, but no easy way of checking fullness.) Very close nearby is generally a kitchen (or more than one[1]) with a meals-only stockpile around it (I often do allow barrels here) that I give a 'make roasts' continuous job as gets given the fat-processing one automagically, and a tannery[2] with a combined skins and leather stockpile (bins often a fraction of the maximum possible) for the obvious reasons. There are other possible arrangements (all the 'raw' meat/fat/skin stockpiles interleaved around the butchers, for example), but skins tend to be fewer so I don't usually give them the same special privilege.
Anyway, when an animal is butchered (knowing this from the struck-down notification, always useful to zoom in on, just in case it
isn't a butchery death!), I immediately go into the t-display of the butchers and set to dump all the inorganics and other waste (bones[3], hooves, horns, nervous system) and check at the same time that my copious haulage forces (dedicated peasants and/or experts in work practices not currently being employed in that) have already set [TSK] against most of the meat-and-offal items in the list. And the skin. If not, there's something wrong, but if at least half are I don't worry too much. The skin might be [TSK] to haul to the stockpile
or to be used by the tanner. I might double-check the combined tanning-force (whether a single guy or a small team) to see if any of them has picked up that job, or if they're all skiving on something else that I might not agree with. Or not bother too much with the immediacy. (I usually get a tonne of skins through trade, though it's always nicer to have 'skin's as well as '(skin)'s in the list.
)
If I have a queue of animals, it's possible that they can get slaughtered quicker than things move out, in which case the total items goes up and the [TSK] count is proportionately less, and I do something to increase the labour. But I usually get this tuned well.
Ironically, having the to-be-slaughtered animals
not next door to the butchers is probably the best factor for getting the throughput working well, the bringing of the new animal delaying until the *CLT* goes down significantly. But with a circa-200 dwarf fort and a pair of butchers for one butchery (with food-hauling alternates enabled on each if I've not got something else to keep them busy) I find no difficulties dealing with the throughput of a surplus cat-n-kitten pasture almost on top of the shop (albeit that I almost only get skulls from that). Obviously that's a more mature fort (6yo, in the case of my current one which I haven't played for the last couple of weeks), but even when it had two or three score dwarves it was also Ok under my particular playing style.
But what I
always do, with the butchery (and some other workshops, like farmer's workshops where I make cheese) is to put them outdoors. Maybe not
actually, outdoors, as in on the original ground level, but often at the bottom of a sky-exposed shaft (with optional retracted bridge at the ground level, or above some 'chimney-like' walls built to prevent 'archer-overlook', in case of the possibility of aerial attack or scavenger-type birds[4]). Even if there is a problem with rotting, you never get miasma under an open sky. Wasted meat/etc, of course, but in my particular style of fastidious micromanagement I don't need to spot purple mist to know that something has started to rot. I aim, of course, to ensure that it never gets to that stage.
[1] Gets awkward with auto-generated fat-processing jobs
[2] Ditto for more than one tannery, albeit that you get a better opportunity for throughput.
[3] I tend to de-forbid the bones that have been dumped in the Z-Stocks menu, indicating that they're now in the dump. Another open-to-sky location where rottable stuff like vermin remains, nervous system and random bits and pieces of beings that
used to be my enemies find themselves, in amongst the megatonnes of raw stones cleared out from rooms and corridors and soon-to-be-flooded/magmaed areas that I like to make neat-looking before I set the liquid flowing in...
[4] Vultures particularly like cheese, IIRC. Go figure.