Hi, this looks like the close enough one..Gits oilers in that style are available in 1/4-32 thread.
Need to check the pitch of that thread.
McMaster-Carr
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You are spot on. I check the end of the thread and beginning section on the top. Diameters are diff.. end is 6.35 mm and top is 6.5mm. Going with 1/4’’ - 32 shown in the picture attached. This has a thread length of 5/32 which is approximately 4 mm.. and the thread nominal diameter 6.35 mm.. guess it will be a trial and error.. I could not find Git mfg cups at McMaster. They have generic one manufactured in Mexico and it’s $20 a piece..Many old english lathes were imperial.. and your 6.63 falls around the .25" .261" it is 10 thou over, but it might be tapered so it doesn't leak.. I don't know.. if it's .25 at the bottom and .261 at the top then it probably is slightly tapered.
Correction, 5/16"=8mmThis has a thread length of 5/16’’ which is approximately 4 mm....
Sorry, I wanted to type 5/32 which is 4mmCorrection, 5/16"=8mm