As Craig mentions there are a lot of options beyond Echo. 3 routes up to inspiration point (and three routes back) making for 9 different semi-loops. Combine the hike with an overnight stay at Mt. Lowe trail camp to test your gear if needed.
Henninger Flats is short but steep and you can continue on to Mt. Wilson to get your heart going or you could drop into Idlehour Trail camp for a lunch by the year round creek.
Chantry flats offers a variety of hikes of various lengths and difficulty. The upper Winter Creek trail to Mt. Wilson and return on the Sturtevant trail(about 16 miles).
The Mt. Wilson trail out of Sierra Madre to Mt. Wilson and back is another.
Further out, past your 20 minute limit but not by much is the Mt. Baldy area with a variety of trails, then further is the Lytle Creek area.
If you really want trailess adventures just step off any trail and start bushwacking through the poison oak, poddle dog bush and rattlesnake dens. I'm too old for that stuff anymore.:-)
I would suggest getting the Harrison Angles Front Country Trail Map and the Angeles High County Trail Map.
Hey Bob, I was on that Sheep count (up Stoddard Canyon). Sorry I didn't now you were part or it or I would have hunted you down.