Praise be to Allah.
Adhkar al-sabah (supplications recited in the morning) are not connected only to the time when the sun rises; in Arabic the word subh (morning) applies to dawn and the first part of the day, as is stated in al-Qamoos al-Muheet, 291 [a famous Arabic language dictionary – Translator]. Allah says (interpretation of the meaning):
“… and celebrate the praises of your Lord in the evening and in the morning.”
[Ghafir 40:55 – Yusuf ‘Ali’s translation]
Ibn Katheer said in his Tafseer (exegesis) (4/86) that this means in the early part of the day and in the early part of the night. But the best time of the day for making dhikr (reciting words of remembrance) is after the dawn prayer, because Allah says (interpretation of the meaning):
“… and glorify the praises of your Lord before the rising of the sun, and before its setting…”
And the Prophet (peace and blessings of Allah be upon him) said: “Whoever prays Fajr in congregation, then sits and remembers Allah (makes dhikr) until the sun rises, then prays two rak’ahs (units of prayer), will have a reward like that for Hajj and ‘Umrah (major and minor Pilgrimage), complete, complete, complete.” (Saheeh al-Jami’, 6222).
Based on the above, this is how we understand the reports narrated on this topic, such as the hadeeths (narrations):
“Whoever says, in the morning and the evening, ‘Subhan Allahi wa bi hamdih (Glory and praise be to Allah)’ one hundred times, will not come with anything better than it on the Day of Resurrection, except for one who does likewise or does more.” (Reported by Muslim, 2692).
“Allahumma bika asbahna wa bika amsayna wa bika nahya wa bika namoot wa ilayka al-nushoor (O Allah, by Your blessings we reach the morning, by Your blessings we reach the evening, by Your blessings we live, by Your blessings we die, and unto You is the Resurrection).” (Reported by Abu Dawood in al-Sunan, no. (5068); al-Nasai in al-Sunan al-Kubra (9836). Al-Albani said it is a saheeh hadeeth/authentic narration).
Therefore, you can say adhkar al-sabah when you wake up from sleeping, because it is still the time of subh (morning). And Allah is the Source of strength.
Note: Muslim reported in his Saheeh (no. 747) that the Prophet (peace and blessings of Allah be upon him) said: “Whoever sleeps without doing his regular dhikr or part of it, then recites it during the time between the Fajr prayer and Zuhr prayer, it will be recorded as if he had recited it at night.”
In Sharh Muslim (6/27), al-Nawawi said:
“This indicates that it is mustahabb (liked, preferred), to recite awrad (dhikr, supplications) regularly, and that if one misses it, one can make it up later.”
In ‘Awn al-Ma’bood (4/198) it says: “This hadeeth indicates that it is permissible to recite wird regularly at night, and that it is permissible to make it up if one misses it because of sleeping or any other reason. It also indicates that the one who does this between the Fajr prayer and Zuhr prayer is like the one who does it at night, so it is mustahabb to make up tahajjud if one misses it at night.”
And Allah knows best.