Old Circline Task Lamp Transformer

I have a very old Circline lamp - it is one that has a magnifier in it and don't want to spend the additional cost to replace it with a new lamp. It is a single circle bulb (see below). There is an in-line transformer that says: Robertson Transformer Co" Cat #PT-22 for one 22W Circline Lamp" It is .39 amp. 118 volts Blue Island Il. The transformer gets very very hot, and I wanted to find out if there is some other type of transformer that I can use that may be more efficient. It uses a GE Cool White FC8T9-CW Rapid Start bulb.

Note that the transformed is in line with the plug before it goes into the lamp base. Then there is a switch near the bulb that you have to hold down to start up the lamp. It is some type of starter switch. There is no electrical circuit near the switch or the lamp.

Thanks.