@Robin Searle Glad to hear you had such a good experience the first time around with your tenant. I personally like open houses because it minimizes my time. I can show multiple people the house and be done in an hour or so. I usually give out applications and an info sheet and they can fill it out onsite or at home and email it to me. I use mysmartmove.com and set it so that the tenant pays for the fee themselves. That way all I have to do if there is multiple is review the report before moving forward to calling employers and past landlords. If I get multiple applicants, I look through their application and make sure they are not automatically disqualified (no evictions in the past 7 years, 3x the rent in income, no pets on my insurance's "no pet" list ect). If they look good then I will send them the request to process there background/eviction check. I don't let them do a background/eviction check (costs them money) if I already know they will not qualify so that I can at least sleep at night.
I don't think you should limit yourself to only military. It is a pain to deal with the potential headache of multiple applicants and open houses, but that 3-4 hours of stress once is way better to me than a year or more of headaches because a bad tenant was chosen.
What you might want to do is ask your current tenant if she knows anyone that is looking for a place and let your tenant market for you at the military base she is working. Obviously they will go through the same screening process as any other tenant but you might not even have to advertise if your current tenant brings you the next tenant.
Finally, if I know I have multiple people interested that meet my criteria, I always try to get them all to submit an application and complete background check so that I have multiple applicants to choose from. I always keep track of who submits and application first but the nice thing about mysmartmove.com is that they give an actual "renter" number so it makes it very easy for me to not have to deal with discrimination issues because I pick the "Most Qualified" tenant which is documented by mysmartmove.com's reports.