Fractional-reserve banks supply credit in two ways: they intermediate the savings of their customers and they credit credit out of thin air by issuing fiduciary media, i.e., checking account balances of their customers that are not backed by a corresponding reserve of cash.
There is no “money-multiplier” for the savings of people that banks intermediate into the credit markets. The “money multiplier” refers only to fiduciary media that banks create out of thin air on the basis of their cash reserves. If the Fed provides prints more money and buys securities from banks, then the banks banks can create a multiple amount of checkable deposits on top of those addition cash reserves.
A monetary system with a central bank and fractional-reserve banks is inherently inflationary, as you say.
Take a look at Murray Rothbard’s book, the Mystery of Banking:’