Many thanks to all who shared your institution's direct mail processes for schools and teachers!
In continuation of paying it forward, here's the summary of responses:
Direct Mail and Teacher Communication Suggestions
Western Heritage Museum + Lea County Cowboy Hall of Fame (New Mexico)
o Use email to reach teachers directly (include administrators, principals, and some school staff.)
o Distribute flyers to schools.
Meadows Museum – Southern Methodist University (Dallas, Texas)
o Mail classroom poster to schools every August, addressed to Art Teacher, Principal, and Spanish Teacher, not by name (Programs on one side, collection image on the other).
Whatcom Museum (Bellingham, Washington)
o Drop off boxes of a brochure/booklet to district administration buildings, and have them distribute to every teacher via inter-office mail.
o Distribute PDF flyer to teachers directly via email.
o Send printed flyer at start of the year and digital version at start of Spring semester.
o Host a teacher fair early in Fall semester to distribute information (offer snacks/alcohol).
Hawaii State Art Museum (Honolulu, Hawaii)
o Program coordinator drops off event flyers and brochures on school visits.
o Send direct mail and emails to school principals rather than individual teachers.
o Use targeted social media ads.
Slater Memorial Museum (Norwich, Connecticut)
o Drop off brochures, etc. at the local teachers' union office and have them distribute to their members. Helps if you have a personal connection with someone there.
The Museum of Flight (Seattle, Washington)
o Send an electronic version of materials to a district administrator for approval, then produce the requested number of printed copies. Drop off the materials at the district office and have them distribute to individual teachers.
o Get materials to PTA members/leaders, who can deliver to the proper person/people at their individual schools. Might help to exhibit at local and state-wide PTA conferences.
Arts marketer (Shaker Heights, Ohio)
o Send relevant emails to teacher lists segmented by grade level, subject area, etc.
o Get teacher email addresses from district-level arts curriculum specialists.
o Add an email signup form to your website to build your email list(s).
o Find out about professional development events offered by districts and try to get your materials included in the resources/goodies that are handed out.
o Offer raffle prizes to teachers who submit their contact info at events. Raffle prizes that aren't specifically student-focused are more popular (i.e. free exhibition tickets vs. a free field trip).
o Use paid ad campaigns on social media.
Editorial and educational consultant (Sherman Oaks, California)
Host events like Tres Museos: Museum Day for Educators in San Antonio