Crafts

To clarify, does the second pair have a channel through which the elastic runs freely, as if the elastic was a kind of drawstring that doesn't need to be tied? If so, you can open the seam and run another piece of elastic through, just longer, and then either securely stitch or knot the ends.

But I guessing that pair is not constructed that way, but rather that the elastic is sewn through the waistband -- so the surface looks ridged kind of like ruffles potato chips? If so, it may be possible to choose two places around the waistband, make tacking stitches to secure the elastic on both sides of where you are want to add, say, an inch on each hip. Then you can carefully snip the elastic; you will probably need to open up a little bit of the inside wait and to access the elastic, to snip it without cutting the fabric. Then sew that little access place closed again.

Essentially what you are doing is taking pants which are elasticized all the way around, and turning them into pants where the back is elastic, and the front is, but there is a bit on either side which is not stretchy. You see this sometimes in old guy pants which have short segments of elastic but just to add a little "give." Those pants usually also have belt loops, to make them look less casual.

Again, I may have misunderstood the structure of your pants. Regardless, I agree with others that you shouldn't just swap out new elastic for old, because that's often what fails first. You can buy elastic in packages, various widths and colors. You also might be able to convert your first pair of pants so that the back half is elasticized and the front half is drawstring; I've seen pants that were designed that way.

Good luck!