Origami has long been renowned as a simple yet creative form of art and its folding techniques have recently inspired advances in design and fabrication of miniature robots. In this work, we present the design and fabrication novelties, enhancements, and performance improvements on MinIAQ (Miniature Independently Actuated-legged Quadruped), an origami-inspired, foldable, miniature quadruped robot with individually actuated legs. The resulting robot, MinIAQ-II, has a trajectory-optimized leg actuation mechanism with longer stride, improved traction, less flexure joint bending, and smaller leg lift resulting in faster and smoother walking, better maneuverability, and higher durability and joint life. In order to maximize the joint fatigue life while keeping the leg design simple, the initial four-bar mechanism is optimized by manipulating the joint locations and changing the leg link into a non-straight knee shape with a fixed-angle lock. Despite having a 1 cm longer frame to embed its new actuation mechanism, the overall weight and dimensions are similar to its first version as its legs are no longer extended beyond its frame. As a result, MinIAQ-II is 12-cm-long, 6-cm-wide, 4.5-cm-high and weighs 23 grams. The test results demonstrate the improvement in speed over its predecessor from 0.65 to more than 0.8 bodylengths/s at 3 Hz, and an approximate decrease in body's roll ±21° to ±9° and pitch from 0°-11° to 0°-7°. The independent actuation and control over each leg enables such a robot to be used for gait studies in miniature scale, as is the next direction in our research.