I don't think expanding the court is a winning move long term.
Short term, we need to pass legislation protecting a woman's right to choose. The majority of Americans would agree with this type of legislation. Same with marriage equality. Also, explicitly pass legislation adding protection to LGBTQ to the existing anti-discrimination laws. DON'T rely on a SC interpretation of the existing law -- make it explicit. This isn't all 100% safe, because it could be undone by more legislation, but it's less risky than leaving these things up to SC precedent right now. These are all things the majority of American's agree with. This is how Obama rectified the Lilly Leadbetter ruling.
Longer term, we need to do something about the Republican's advantage in the electoral college, and even bigger advantage in the senate. The court more conservative than the American public because the senate is tilted in favor of the republicans. This needs to be fixed before expanding the court will be anything other than a short-term fix.