The Picasso Museum is having a special exhibit on the Guernica. The painting itself remains in Madrid, but they have a lot of information about its history, works inspired by it, etc. If that is of interest, you will want to see this exhibit. Covered by Museum Pass.
I second the recommendation of the Petit Palais - not only free, but not too crowded when I went, with some very good pieces.
The Rodin Museum was very crowded when I went last month (not "Impressionists at the Orsay" crowded, but still not pleasant). However, it is very good, and the gardens were less crowded than the interior. Covered by Museum Pass.
I wasn't taken with the Army Museum, as it's a lot of reading (felt like a textbook more than a museum). But, if this is an area of interest, you may like it better than I did. Covered by Museum Pass.
The Nissim Camondo museum is very interesting, and is covered by the Museum Pass.
The Orangerie not only has the Water Lily Rooms and the Walter Guillaume collection, but right now is having a special exhibit on Monet's influence on Abstract Expressionism. They have a few later Monets on loan from the Marmottan, as well as some great stuff by Helen Frankenthaler, Jackson Pollack, and others. It was also interesting to learn just how vitriolic the initial reaction to the Water Lilies was; the Abstract Expressionists were being quite "avant garde" by championing Monet when they did. Both the regular and the special exhibit are covered by the Museum Pass.
Not a standard museum, but the Gobelins Manufacture is having an exhibit on 100 years of tapestries. What's fascinating is that as techniques advanced and painters started making designs for tapestries, the tapestries look like paint rather than thread. Even up close, they are able to simulate the appearance of a watercolor wash, or (even more amazing) the look of paint put on with a palette knife. Totally fascinating, and as a benefit totally uncrowded. The only catch is that they don't have a bag check, and if you have a backpack you must wear it on the front for the whole museum (not too comfortable for me). Entry is €8, and it's not covered by the Museum Pass.
I haven't been in about 10 years, and it seems they've expanded since then, but I liked the Museum of Counterfeits (Musée de la Contrefaçon). It not only has fake purses and such, but more sinister things like fake car parts and fake medicines, that have caused fatalities. As a bonus, it's near the Porte Dauphine metro, which has one of the few intact glass metro canopies. Not on the Museum Pass; entry €6; limited hours (website says Monday to Saturday, 14:00 to 17:30).
Note that at all museums, expect security checks. These will vary - metal detector, x-ray, hand-wanding, or some combination of these are all possible. They do take time, so allow for this.