Suppose the lowest offer on stock S is $10. This offer is not the market maker's offer. A trader places a market order to buy stock S on the NYSE. Can the NYSE market maker "intercept" this market order and sell stock S at $9.99 using the market maker's own inventory, bypassing the lowest existing offer?
A market maker can change his bid price to match the ask, but the trade can only happen when an ask price matches his bid price. That's the nature of the order-matching systems used. The prices you see are only what the price of the stock was at last trade, but that doesn't mean one or more market makers haven't adjusted their bid prices either down or up from there in the meantime. So you can have quotes that are either higher or lower than bid and ask.