You can use the private key for the same wallet. The private key does not change with every transaction. It is permanent and stable. If you want a new private key, then you need to create a new Bitcoin wallet.
Your terminology is wrong.
A "wallet" is a container that
holds one or more "private keys" (and the associated "addresses"). A "wallet" is not a "private key" or an "address".
If you want a new address, the wallet will create one for you by generating a new private key. Most wallets do that automatically every time you use an address.
Typically, a wallet has a unique "seed" or "recovery phrase", and if you want a new wallet, you get a new seed.
People write things like "send bitcoins to a wallet" or "a wallet has X bitcoins" for convenience when they actually mean "send bitcoins to an address in a wallet" or "a wallet has addresses with a total of X bitcoins"