This may sound like a strange question, BUT - last January I was personally invited/recruited by a department chair to apply for a position in a newly formed unit at an Ivy. During the search, I was actively recruited, e.g. the chair kept in contact weekly, worked to sell me on the position and institution, etc. I advanced to the finalist round and was invited to a campus visit, but ultimately was not offered the position (I lost during the faculty vote by one vote). When the chair reached out to tell me, he expressed disappointment because I was the person he wanted to hire. He has since made an effort to keep in touch via email about research topics, etc. Yesterday, I received a handwritten letter thanking me again for my campus visit (which took place months ago, in April) and insisting he'd be happy to support my work in any way he can.
My question is - can/should I ask this person for a letter of recommendation? I know it seems weird since he was on a search committee that ultimately didn't select me for a job, but also I am a junior scholar deeply unhappy at my institution and on the job market this year, and his institutional affiliation and clout the field could be helpful for my prospects.