Here is an idea for light enough weights.

I have a pair of five pound ankle weights. Each one has 10 compartments for a 1/2 pound iron bar, which come with the set. If you can find these same ankle weights, you then have the material to begin exercising with weights as light as 1/2 pound. You could use the ankle weight unit with from one to ten bars in it.

Also at a gym, you could take your own add on weights in small increments. One manufacturer of them is Plate-Mate, which makes magnetic add-on weights. Then you can make smaller, safer weight changes in your strength program.