When an animation is applied it overwrites all the bones' SRT. Storing them all so you can apply them, then another animation would be a pain. Instead, just apply the old animation with a time >= the last keyframe. The easiest way to do this is to set the duration on the animation to something high, like int max or 9999. Then use AnimationState to do the mixing.
firstAnim.duration = 9999;
state.setAnimation(0, firstAnim, false);
state.addAnimation(0, secondAnim, false, 2); // Need to set delay, can't use 0.
The delay has to be set for the queued animation, can't use zero which would try to use the first animations duration as the delay.