I am assuming that the "odd hits" are hits received that have not been enough to remove a stand of the unit which is carrying the banner - similar to the end of the shooting phase and any odd, left over hits are discarded.
So therefore, at the end of the combat round, the "odd hits" are discarded and don't carry over with the unit for any future combat and this is when the effects of the banner are removed i.e. the extra hit per stand.
For example:
A unit has 3 stands with 3 hits per stand.
It is also carrying the Banner of Fortitude.
This makes the unit have 4 hits per stand until the end of it's first combat engagement for the battle.
The unit is attacked and takes 7 hits total. Due to each stand now having 4 hits each a single stand is removed as a casualty and 3 hits remain.
There is no further combat - no pursuit or combats that result from advances etc.
The unit has lost 1 stand and has 3 remaining "odd hits". These "odd hits" are discarded and the banner becomes defunct. If the remaining "odd hits" weren't discarded but the banner becomes defunct, the hits per stand now drop back to 3 per stand and another stand would become a casualty. Hence, the removal of the "odd hits" before the effects of the banner are removed.
I hope that is the way it should work. Please correct me if I'm wrong.
