If the hairspring is gummed up or damaged, you'll need to send it in to Starrett. These old back plunger indicators are very hard to work on.
With what Starrett charges for repair, you may want to consider a new Mitutoyo back plunger. Much better and parts are available.
According to Starrett's repair guide, you have to make a special tool to work on these. Oil is generally not a good thing for indicators since they can ruin the delicate hairspring. Special watch oil is used mainly on the jeweled bearings only.