The link reached by clicking this post's title gives great strategies for overcoming the perception that you are "overqualified" for a lower-level or similar-level job within your current industry.
Some of these tips are also useful if you want to transition to the non-profit field from the for-profit field, or switch industries. In these instances, it is easier to make the case for you starting at a lower level regardless of your past experience and skills.
In my experience as a CEO and manager, attitude is the biggest red flag when an employer is considering a highly experienced candidate. Will that person be happy in a job reporting to someone less experienced? Will that person stay in the job for more than six months? Or are we simply a stepping stone for their next job? You'll stand a better chance of being considered and hired if you can honestly commit to staying one to two years, and convince the hiring person/people of your humility and sincerity.
Be honest with yourself FIRST, though. Don't tell people what they want to hear and then renege on your promise. That's out of integrity, and poison if you are trying to enter a new field or industry. Word can spread that you are unreliable, and instead of being a stepping stone, this first position could be your last. Only commit to what you know you'll deliver.