Hi @Jeff S.
Great topic, by the way. Although when we started investing, we did all the work ourselves, as soon as we had 3 properties under our belts, we began hiring out the work. First was HVAC, then plumbing and electric--I do not leave those items to a handyman. I use the very same plumber and electrician for every single project--big or small. They've gotten to know me and how I like things done (right, the first time, and to code, please!). Maybe I pay a little more, but the peace of mind in knowing the job is done right is worth it to me. Plumbing, and AC comprise the majority of the trade-specific repair calls I get (until, of course, my trades get in there and fix everything right--then the phone is quiet).
As for the little piddly stuff, it is not easy to find a jack-of-all trades. My suggestion is to be involved in your local REIA--ask around. Be specific. Beg for the name and number of everyone's favorite handyman. That's how I found the guy I use. Most of the time, I can simply call him up and tell him what to fix. He sends me pictures of the before and after, and calls if there are any surprises. I've been working with him for nearly 6 years, so it's a smooth process. I'm probably going to lose him soon, as I've given his name out to so many people, he is now often too busy!
So, I have learned to 'batch' projects for him. I'll tell tenants if a repair is not urgent in my opinion, and that my repairman will be around to fix things next time he's in the area. When my handyman is available, I make sure I can keep him busy for at least a half day, but a full day if possible. I've learned to give him a heads-up, that I have a few things here and there that need attention. He works out his schedule and squeezes me in.
This past year, I just began something new, that I think is going to work out great. I asked all my tenants to email me a list of any repairs or concerns that needed my attention. I had my handyman go with me to each of our properties (at the time, 14), and we inspected everything together. He made most repairs right on the spot; others, we scheduled them to be fixed at a different time, when we had the materials or such. If something needed a different trade, I scheduled that right there in front of the tenant. Tenants loved this! And I managed to keep my handyman busy for two full days. I paid him more than the $375/day he usually charges, because he is such a thorough and fast worker. He's worth his weight in gold. I think I'll keep doing this, probably twice yearly; though now we have 18 properties, so it'll take three days probably. I mentioned this because it might help you with your situation. If you're pro-active with repairs, you can keep ahead of them and schedule them when it's more convenient for you. Of course, we can't catch everything--in those cases, try to batch the repairs to make it worth a good handyman's time.
BTW, even with 18 rentals, I can't keep a handyman busy anywhere near full-time. If I don't have a rehab project going on, I use him, on average, 2-3 times monthly.
Wow. Sorry for the lengthy comment. Hope it helps. :)