How deep the sleep is depends on the sleep cycles. This overview from shows, that the deepest sleep stages 3 & 4 only happen during the first half of the night. The second half is mainly REM sleep. However our brain is highly active during al these cycles.
Image source: wikipedia
Because even at night there are noises and plently of stimulants for the brain, it needs a gatekeeper which decides which stimulant to get through. This is the job of the thalamus. So if you hear your alarm or if you don't depends wether the thalamus "decides" that it is an important noise and therfore should be recognized by the cortex (2,3)
One example of that gatekeeper function are Mothers with newborn children. If their child does tiny noises they will wake up, no matter which sleep cycle they are in, because the thalamus recognizes the child as important. On the other hand cars which are passing by on the outside street, which are even louder that noises from the child will not wake up the mother.
To answer the question: most likley your alarm will not be as important to you as your newborn child, but if it's important for you to wake up with your alarm, you will even with 30% less noise, no matter of your sleep stage. However disorientation and headaches will not get less. Probably you are in a deeper sleep cycle when your alarm goes of, to change that you should alter the time you go to bed. A roughly estimate is, that you need 90 minutes to get one full cycle of sleep. So try to get 6:00 or 7:30 or 9:00 hours of sleep.
Read more about sleep hygiene here or in the link of Adamawesome4.