I have a sensitive drilling attachment (SDA) that I bought mainly for my drill press. For those not familiar with them, the smaller chuck is on a spring loaded shaft, so instead of using the quill on the DP, the knurled collar (that has a bearing inside) can be pulled down - much better for 'feel'. The chuck on mine goes up to 1/8", with a 1/2" shaft to go in the driven chuck.
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Today I had to drill some 1mm holes and my battery drill chuck would not hold them, so I tried putting in the SDA and used that. Not only did it do the job very well, but by hanging onto the collar, I was able to prevent the drill wandering around as much.

If you have a SDA then here is another use for it, but if you have not, mounting a bearing on a chuck/ shaft similar to this (effectively a SDA without the telescopic spring bit) will at least help get better alignment when starting a small hole, as well as support the weight of the drill a little (smoother feed, less likely to feed too much at once).

Michael