These hanging closet rods that hook over the top rod, take only minutes to put together. Most are adjustable in width, some in length also.
Ive done similar in all my closets. First I removed the shelf, and moved the existing rod higher. Just had to move screws from both rod holders and relocate giving room for hangars to still easily get over the rod. Then I bought a new rod kit at Lowe’s, my closets are standard 48”, and installed to correct height. Make sure it’s low enough so the clothes on top rod don’t touch bottom rod and clothes on bottom don’t drag the floor. 😀
easy peasy... Buy a clothes pole and brackets. The brackets install with a screw. This kind of pole holds better than anything else.
I will add to the two answers above. Think about placing the top pole and shelf as high as you can reach leaving more shelf space above the bottom pole. You might find that having shelf space here is more convenient than a taller space at the top.
Agree with Rymea about making the top bar as high as it can go while still being able to reach to hang up a hanger. This tip is called 'raise the bar'.