Can you tell us where the run out is coming from? I know that sounds facetious but you are reporting 0.004" TIR on the shank of an end mill and there are numerous interfaces in between the spindle and that end mill.
- Is this an integral chuck or is it a chuck with a Jacobs taper installed on an R8 arbor? This is two interfaces.
- What is the run out inside your spindle taper, taken at and averaged over three separate spots in the taper? This is your baseline and without it you cannot attribute any deviation to a thing inserted in the spindle taper. This is another interface.
- And then there is the effect of clamping something in a 3 jawed thing. This is the interface with the biggest potential for run out.
A drill chuck typically has 3 jaws. The way to look at this is like a 3 jaw chuck on your lathe. As we all know, putting something that is precision ground into a 3 jaw chuck will cause substantial run out simply because it is a second operation chucking. In order to have that precision part run true you must either dial it in with a 4 jaw independent chuck if accuracy is important or use a collet if accuracy must be close and fast.
Due to the nature of the beast, a drill chuck will tighten variably because it relies on a threaded spindle. Depending on the diameter of the thing being clamped, the jaws will locate on that threaded spindle at different spots and those spots will differ every time you clamp something in the jaws. The same thing happens with a lathe scroll chuck; different spots in the scroll causes the jaws to run out like crazy.
So, expecting low run out on a drill chuck is expecting a lot just based on the mechanics of the thing. Throw in all those interfaces and you have an even greater amount of uncertainty.
The other thing to understand about drill chucks is that not all chucks are made equal. The jaws of a 1/2" Albrecht chuck are ground and hardened. Each jaw is 0.195" thick and are accurately machined. The jaw guide is likewise machined and hardened and then ground. The spindle is a precision ground Acme leadscrew and the head of that spindle has slots to match the machined contours of each jaw. When the chuck is closed on a round thing, everything moves pretty precisely and repeatedly but even an Albrecht chuck will run out due to the nature of a 3 jawed thing; it has a run out of less than 0.04mm/0.0016".
Sorry, didn't actually intend to go off like this but drill chuck accuracy, while important, has to be tempered with the knowledge that a drill is not a precision tool. Just the drill alone will add thousands of inches of run out all by itself so if the chuck is anywhere close to 0.002 TIR (and you're sure its the chuck and not something else) then I wouldn't be too concerned about it.
