I gather the part is 38mm in diameter Michael but it would help to know the length of the part and the size of the hole through it. Also what material did you start out with (was it already cored/drilled or solid for instance?
If it has a small (relative to the diameter) through hole then the approach you originally mentioned might well be the best one. The need to provide tailstock support would depend on how far it extends from the chuck, although if in any doubt then of course support it. Small cuts are the order of the day (with a sharp tool).
However, it may also be possible that thinking through the machining operations before starting might have helped you avoid the problem. You may well have done this of course but the reason I mention this is that I have often found myself in a (self-imposed) bind because I didn't think the whole process through to begin with.
For instance, starting from the solid, could you have turned the work between centres before drilling/boring it? Maybe not – but this might have been easier than making a mandrel or other fixture. Alternatively – with a pre-cored piece, could you have filled the holes at each end and then centre drilled them (for between centres turning). I don't know the answers in this case but you may want to think about it.
I should mention that I do have some commercial mandrels (acquired at a boot-sale) and have only used one of them (once). They would be very useful if I was machining volume parts in some circumstances but most of my work is 'one-off' from stock material. Having dug myself into holes in the past, I now try to think through the work holding required before jumping in. For instance, I try to keep the work part of the parent material for as long as possible and allow extra length if I'm turning between centres (parting/turning off surplus later) .
You may well have done all this of course (I'm not suggesting you didn't) but the advice is good but easily forgotten and I still make silly mistakes in this area even now – which is why I mention it. 
Regards,
IanT
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