You have a couple of good reasons to write the 3q note.
- Keep in touch with your ex-boss, who apparently spoke good of you. You may need his/her reference again and again in the years to come.
- What's the problem of letting people know that you don't have a job yet? Being unemployed is not a crime. In fact you should let all of your friends know that you are looking for job now, therefore your friends may offer you some help when they got chance.