Delta hedging a long call after a sharp stock rally

You own a European call and want to keep the position hedged continuously against small moves in the underlying. What hedge do you put on, and how must you adjust it if the stock price suddenly jumps up?

Short $\Delta$ shares per call, and short more stock after a rise. Holding $-\Delta$ shares makes the book insensitive to small moves. A long call has positive gamma, so when the stock rallies its delta increases and you must sell additional shares; after a fall you buy some back. This sell-high, buy-low rebalancing is how a long-gamma position earns back the time decay it pays.