When I made my tiny engine I made use of a commercially bought slender rod turning steady I gather from some of the comments that these are not easily acquired so I decided to make one:-
Made from standard stock size bar with a 1/4" Square HSS tool
It manufacture is quite simple so I won't go into to many details.
To ensure that the bush was on the centre with the tool
I first measured the tool height and stamped it on the steady so I could get back after a tool grind

I then carefully set the tool on centre height and on centre.

Locked the cross slide up and with tools in the head stock drilled out for the bush.

All that remained was to make a support bush.
And heir's a video of it in action
its a real handy little tool
Stew