There's a major bread problem up here in the Valley (and really in many parts of LA). You're on the right track with Jon's, but the best solution I've found is Victory Bread on Victory and Hazeltine.
It's an Armenian bakery. Tiny storefront that usually has a couple dudes out front with cigarettes and small cups of strong sweet coffee. They make Armenian barbri, puri and matnakash, as well as lavash, and they also do nice loaves of rye and whole wheat bread that they'll slice for you. Everything I've ever bought there has been handed to me still warm. I now swing by several times a week for sandwich bread for my kids' lunches, but when I want good bread for good cheese or homemade soup I'll also get the aforementioned barbri, puri or matnakash. I don't think of myself as a bread person, but to me these are fabulous--on par with the best stuff I've gotten from high-quality "White People" bakeries--crispy edges, soft insides, a nice salty tangy flavor that makes them more than just bread. They're open 7 days and appear to sell their inventory at the many Armenian markets in the area, but there's nothing like getting it fresh and warm.
Cheap too--$3 for an Armenian loaf worthy of eating on its own, $2.50 for the sliced sandwich loaves.