Multiple suicidal gunshot wounds are rare but by no means unknown, especially where the target area is not the head. Similarly, a reluctance to disfigure the face and head is a known reason for choosing other target areas, usually chest, then abdomen. I take these facts from memory based on authoritative sources, as any firearm suicide cases of which I have had close knowledge have been single headshot.
Giving a calibre of 5.56mm points towards you using something like an AR15, because sporting, hunting and ranch rifles in this calibre in US and UK are generally referred to in Imperial measurement of .223", though the designation of firearm calibres is a haphazard procedure, to say the very least.
If you are using an AR15 or a clone you will have the option of semi auto operation at least, which should facilitate a follow up shot. 5.56mm bullets cause devastating injury, whether NATO full metal jacket (which tumble and fragment along the wound track) or expanding .223" . Like other high velocity rounds , expanding or not, these cause a temporary cavity along the wound track which can be ten times the diameter of the permanent track. This temporary cavity collapses within milliseconds but the damage is done. Organs and structures which lie within this stretched area are likely to be damaged or destroyed even if not directly struck by the bullet. Encapsulated organs, like heart, liver, kidneys, are particularly susceptible to this sort of damage. I can only think that a person already suffering a chest wound from such a round would have great difficulty in operating a bolt action rifle and then resetting the muzzle towards her/himself for a follow up.
A self inflicted head wound using this sort of cartridge would possibly/probably cause a so-called Kronlein shot, where the brain is completely eviscerated through large defects in the skull, so fear of ugly mutilation may be well founded.
Over the turn of the 20th/21st centuries four young recruit soldiers were shot dead in separate incidents in mysterious circumstances in the same army camp in southern England. Their deaths were initially presented as suicide, or killing themselves (not necessarily the same thing to a UK Coroner) but in at least one case an open verdict was returned. They used their issue rifles which are chambered for the 5.56 NATO round. Two had a single GSW to the head, but a third had two to the head. The fourth had five chest wounds, and I can only imagine that full auto fire was selected, and the poor lad managed to depress the trigger long enough to get a burst into his chest.
Are you Geoffrey Chaucer ?