In parliamentary systems like in the UK and Israel, the minute a new Government is formed there is an official head of the opposition. It's not the office that counts though, as much as that there is all the time a counter to the government that presents to the public the alternative or what actions they think should be taken.
In the US you don't have anyone who tries to present a real alternative and opposition to the president. You obviously have two opposing forces in the Congress but that's not the same. The party that lost the presidential race should nominate their sort of anti-president to head the opposition to the president. Then before the next elections there can still be primaries for the nomination.