Since getting a set of R8 collets for the mill, I rarely use a chuck any more. I have a 24 pc. set x 1/32" which allows me to use any of the drills with a collet. The advantage is by removing the chuck from the mix, the drill tip is closer to the spindle which gives you more vertical work space. This can be an important factor on a small mill. The collets also have better runout specs. than typical chucks.
R8 collets will draw down 1/32" from nominal size. This gives me the capability of mounting any diameter from 1/16" up to 3/4". Depending upon the width of the slots in the collet, you could have a problem drawing down a 1/16". Even if you could clamp an 1/16" undersized shank, the grip on the shank suffers and runout is increased.
If you elect to buy a set by 1/16", you could fill out the set later but it would most likely require you to buy individual collets as sets by odd multiples of 1/32" are rare, if they exist at all. I made that mistake when I bought my 5C collet set.