If you wish to re-order a Flickr photo in your photostream, you will have to change its post date. To change a photo's post date in Flickr, follow the process below:
Quick Tip to ensure your Photos never go missing
Photos are precious memories and all of us never want to ever lose them to hard disk crashes or missing drives. PicBackMan is the easiest and simplest way to keep your photos safely backed up in one or more online accounts. Simply download PicBackMan (it's free!), register your account, connect to your online store and tell PicBackMan where your photos are - PicBackMan does the rest, automatically. It bulk uploads all photos and keeps looking for new ones and uploads those too. You don't have to ever touch it.