The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are ${sigma}$-rigid ${delta}$-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where ${sigma}$ is a ring endomorphism and ${delta}$ is a ${sigma}$-derivation.