Once you've set up the campaign you can't change its start date. However, you can finish the campaign at any time by changing its status from Active to Finished.
When you finish the campaign it will be transferred to Finished section of campaigns list, and HypeAuditor will stop monitoring influencers for new mentions. You will be able to restart the campaign at anytime. Finished campaigns do not count against your plan limits.
When you restart the campaign HypeAuditor will collect posts with campaign's mentions over the time the campaign was in Finished status. However, please note that historical data for that period with daily engagement, spendings, etc. won't be available.