The only way (in addition to Minderwiz's suggestion of contacting the hospital for the time of birth) is to engage in a rectification of one's life in relation to the birth chart.
This is an arduous undertaking and is better left to the very experienced astrologer, preferably one who has performed numerous rectifications.
The astrologer will ask for anywhere from five to ten MAJOR life events, like date of marriage, birth of children, illness, accidents (of major proportion), profession, etc. Also, one's looks are taken into account (stature, shape of face, weight, etc). From this info, the astrologer goes on an expidition along with the birth chart to ascertain correlation of events to possible angles of the chart, culminating with an appropriate ascendent & degree.
AG