You can make an order at any time. Orders are placed live on the NYSE and NASDAQ, so any order you make outside market hours will be completed when the markets next open. The US share markets open between 9:30am - 4pm Eastern Time, which is:
2:30am - 9am NZT 15th March - 4th April
1:30am - 8am NZT 4th April - 25th September
Note: Any market orders placed while the markets are closed will be completed at the opening share price (the estimate you see in Hatch the closing share price). Share prices can change a bit in 'after-hours trading' because, while everyday investors have to stop trading when the markets close, but big institutions can keep going. If you're keen to keep track of after-hours trading we suggest that you use Yahoo Finance or similar tools.
Want to have certainty over the price you get?
If you want the most certainty possible over the price you pay, you have a couple of options:
Place market orders when the markets are open. If you don't want to stay up all night, place your orders when you wake up before the markets close.
Place limit orders to specify the exact price you want to pay. You may risk the order not being completed if the share price doesn't meet what you enter, but if it does go through, you'll get the exact price you expect (or a better price!)