TBC
As an afterthought, I always turn small diameter objects between centers. I don't know if there is a difference between TBC and a collet chuck. You might try it. I cut out the two step drilling, and drill the larger hole completely into the blank. I have found that the larger bit going deep into the blank is less likely to drift with the grain of the wood.
Not to disagree with the other gentlemen, but I have tried extensively to use steady rests and find them cumbersome both in the set up and getting in the way when going down the length of the blank in one smooth cut with a skew. If you're unsure about using the skew (I don't know what your experience level is) but......If you use the tutorials on this site, and practice (a lot) at using the skew, your experience at turning long thin materials will be much better. Once you start using the skew and 'get it', a light will turn on and you will say "I didn't know it was this easy".