This already exists. 1 bitcoin = 1,000 mBTC = 100,000,000 satoshi. The exchange rate as of writing is about 846 satoshis = 1 USD or 982 satoshis = 1 EUR. The current usage is thusly:

• Transaction fees on the main network are measured in sats/B (satoshis per [virtual] byte [of transaction size]).
• Transactions on the Lightning Network, a lower-fee instant payment network that runs on top of the ordinary Bitcoin network, are all denominated in respect to satoshis. So a payment on the Lightning Network is traditionally regarded as, for example, 50,000 satoshis rather than 0.5 mBTC or 0.0005 BTC. And the fee on that payment is usually in the range of 50-100 satoshis.
• Goods sold on the Lightning Network are typically priced in satoshis.