If you need it to be up to the second then you will need to restore the next log with recovery, if you can skip the tail of the log or are not bothered about the next log being shipped over, you can just restore the database with recovery as you have previously stated.
Typically you have log shipping setup with a load delay. You would want to roll the database forward to as soon as possible before the problem that occurred that caused you to fail over. If you are simply failing over to do maintenance on the primary then you want all transactions, so apply those logs and do a final WITH RECOVERY restore. If you had a data related event that you want to stop at a specific time, then restore the logs to that point in time then restore WITH RECOVERY.