Before anyone can give you any kind of meaningful advice, you need to provide more detail of what it is you are trying to do or make and what the spindle will be used for, what loading, what working environment etc etc. Otherwise it's just a guessing game.
And how are you measuring to one thousandth of a millimetre? Seems unlikely in the home workshop. Hundredth of a mm is as close as you are realistically likely to get.
But, I am guessing from the 33-1/3 rpm you let slip, that the spindle is something to do with an old record player? In which case it is not critical enough to need to use a floating reamer holder. Yes an 10mm H7 reamer available as cheap as chips online would do the job, held in the tailstock chuck. Ideally you should drill the hole then bore it out to nearly finished size and use the reamer to finish is off. But realistically, drill it to 9.5 mm and ream it should be good enough.
Or you can buy any number of ready made bronze bushes, or sintered bronze self lubricating bushes, that come with a 10mm bore and various ODs and lengths.