Be realistic in your expectations both of the position being applied for. More importantly you must understand that finding a teaching position in a private school takes some planning and effort. If you are not prepared or cannot invest the required
time and effort, then you need to consider other options.
Finding a teaching job is not like searching for a managerial or sales job. Those kinds of jobs in the business world are open throughout the year. Teaching positions on the other hand begin in the late summer or the begging of the academic year and end in the late spring or at the end of the academic year. In order to secure a teaching position for next fall you need to begin the process in November or December at the latest. Contracts are renewed in February and March in most schools. You want to be in a position to move quickly once those vacancies are announced and advertised.
Tailor your cover letter and application to the specific position.
Professionals who have been teaching for a while understand this. You may have even sat in on a committee or two where you reviewed applications for open positions. It's embarrassing to read materials submitted by potentially well-qualified applicants who never make the cut simply because they used a generic cover letter. A generic cover letter or one which you use for several applications makes your work easy but it sends the wrong message about you.
Make sure your qualifications match the requirements as closely as possible. And if, for some reason, they don't match the requirements but you still think you are a good fit for the position, state your reasons very clearly, indeed, compellingly in the cover letter and again in the application. It's a long shot but you just may get lucky particularly if the applicant pool doesn't yield a better match.
Avoid overused phrases and generic language. Words like "dynamic", "creative", "multi-faceted" and so on are best replaced with clear descriptions of you, your personality, your teaching style, your passions and your experience. Read the requirements of the position and determine what the school is looking for in an ideal candidate. Then be that ideal candidate.
This advice applies to just about any job position in a tough economy. Administrators have to make adjustments on the fly. If they know that you will be agreeable and handle ny assignment they give you, that will enhance your chances for being appointed. It will also virtually assure long term employment. Agreeable, competent team players are a huge asset in an academic community. Everybody appreciates those qualities. Students and colleagues alike.
Have excellent references.
No reputable private school will hire you without doing a thorough background check. This usually takes two forms. One is the standard credit and background checks including drug tests. The second is the much more personal references check. Most private schools will ask for the names of three former employers who can speak to your abilities and experience as an education professional. You must ask these people before you put there names on an application. Hopefully you have been smart enough to use them as mentors and advisors in your job searching process. Those referees will be called and interviewed at length. Make sure there are no skeletons in your closet.
Network. Network. Network.
Honestly, you need people who think you are wonderful, gifted, brilliant and so on. Your network should be your fan club. Make it so. Become part of the social media conversation. Don't be outrageous. Just contribute intelligent coments and observations whenever appropriate. Your colleagues will remember and respect you for that.