I usually start my morning run once I get up in the morning.
I have had my morning run since I got up this morning.这句话如果要用完成时的话,就要用since,而不是once.
如果不需要强调完成时,那就用过去时：
I had my morning run after got up this morning.不要用already.
所谓morning run,也是习惯用语,不需要加不加的.