Yes. I'm curious if you can get away with a simple coupler nut. Easy enough to make one for that matter.
First picture is a stock later model Logan dial. Early models had a hex bushing head like your lathe. The hole in the apron bushing is for a pin spanner. The inside of the bushing on a Logan contacts the power cross slide gear. Most of the holes end up on the bottom, but not all. The round bushing/index lubber line is easier to cover then the hex head bushings, but not impossible. Or just thread the end of the extension and replace the factory bushing. Many enjoy making their own dials, Mr. Pete shows making one for a Logan, Winky also made one for a Logan and made a screw extension. For this I've one the |Good| Kinda Quick| Not the Cheapest| route and picked up a Bridgeport mill table dial and carrier. Made a bushing for the dial carrier and it's done. Larger dials on a Logan will be blocked by the cross slide, some just eat the loss of bed width, others make an extension like in the second picture. You will only need a small extension to have screw length for the dial.
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Here's a nice Hardinge dial mounted on a Logan, all be it reversed. It was posted here or maybe the Home Machinist forum. I didn't take good notes. This extension replaced the apron bushing.
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This is what I came up with.
Just need to find my crayons and get it lined up.
I’m getting good’r at this , It even spins true
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