Apply Online & Send your Docs
If you do not meet the criteria, you cannot apply to change your name in the passport. You need to submit evidence of change name on passport.
Step 1: Complete and Submit Form DS-82: Application For A U.S. Passport
Step 2: Submit Your Most Recent U.S. Passport
Step 3: Submit Evidence of change name on passport
Step 4: Pay the Applicable U.S. Government Fee
Step 5: Provide Two Professional Passport Photos
Step 6: Include Two Letters of Authorization
Step 7: Ship Documents to Visa Express
We recommend to do that. U.S. citizens who change their names can travel using a U.S. passport that was issued in their prior name, however you need proof of the name progression, according to U.S. Customs and Border Protection (e.g. marriage certificate, divorce decree or court documents showing the legal name change).
You definetely can travel using the passport shows your maiden name. However, you need to make sure that air tickets using your maiden name and not your married name. You can apply for a passport name change when you return back.