It may be expensive for you to request a different size moving container from the company at the last minute. Your best option is to try and squeeze everything into the container by stacking boxes on top of each other.
Do it safely by putting heavier boxes at the bottom, and then stacking lighter ones on top. Ensure this is done near a corner or the back of the container, so the boxes do not fall down when the container moves.
You could choose to eliminate one or two larger items from the move, such as furniture. Perhaps you have a very old sofa or appliance that would be affordable to replace. Eliminating it from the move would save you a lot of space.