Sighing IS relief! Why is that?
Exhaling is the "resting" phase of the breath. When we breath out the diaphragm moves upward in the body, allowing the muscles attached to the diaphragm to relax. As this happens, the lungs deflate. The Key to getting a full and satisfying exhale is to also allow the ribs to relax, move inward and downward. This will also reduce tension in your back.
Put your hand on your breastbone and sigh. You should feel the chest drop. Try touching the sides of your lower ribs. These ribs should also lower and move inward. One other interesting little fact is that the heart pericardium is attached to your diaphragm! Exhaling decreases pressure on the heart and lowers your heart rate.
If you don't get it out, you will not get much in! By restoring a natural exhale, your inhale will be effortless.