The simple answer is: carry more than one weapon.
Rather than trying to force a weapon to fulfill two separate needs (a ranged weapon like a gun that is also a melee weapon), take two. I mean, if it’s completely necessary and you can’t give up the idea of a ranged weapon that can also be used in melee then you can always take a bayonet. It’s not going to work for taking off the head unless you detach it from the rifle or shotgun but it is an option. A bayonet is essentially a long knife or a short sword depending on how you want to define it that attaches to the barrel and allows the gun to be used as an impromptu melee weapon. It’s not as good as an actual melee weapon (which is why soldiers also carry knives), but it serves it’s intended purpose.
However, when attached to the barrel, the rifle’s use as a melee weapon is limited in terms of motion. The gun isn’t designed for that kind of motion, it can basically slash diagonally and stab. So, if your character needs an actual melee weapon then they should carry a secondary or tertiary weapon to support them at close range. Whether that’s a tool like a machete that can be hidden easily under the coat and will work well for taking off the head or some other kind of silver sword.
I always liked the story in Hunter the Reckoning’s Storyteller’s Handbook about gang members trying to trade in their silver jewelry only to be told that they could actually buy silver bullets. Which is true, your monster hunter actually can purchase them. However, because they were so expensive and because you could never be totally sure, many of the setting’s gangbangers started putting silver bullets as the third round in rather than carrying a full cartridge. The logic was essentially that if the first two didn’t work then the third probably would. (Except it wouldn’t always as the setting had monsters with similar powers and different weaknesses such as the WereCrocs who were weak to gold rather than silver and mages with points in Life and willing to risk the backlash, Gangrel or Vampires with Protean, etc.)
While I get the appeal of a “signature weapon”, the truth is that most people simply carry more than one weapon as you’re bound to encounter different situations/scenarios and no single weapon will ever have an advantage in all of them. Try to limit yourself to about four, but in this case this character might carry as their loadout:
A rifle – for long to mid-range, especially when outdoors for picking off enemies at a distance.
A shotgun – for close range and because you can load it with a variety of different types of ammunition from buckshot, to needles, to turning it into an impromptu flamethrower. My personal opinion is that the versatility of the shotgun lends itself really well to monster hunting in particular because you can load it up with all sorts of stuff and it’s very damaging at close range.
A pistol – the pistol is just a good middle of the road weapon, it’s small enough to conceal on your person, and it’ll have more of an advantage indoors than the rifle and even the shotgun. If you’re in a situation with very cramped quarters, then the pistol is your friend as it’s unlikely to get caught on the environment in the way a longer weapon will. Depending on the type of pistol, you can load it up with different kinds of ammunition.
A short sword/knife/sword – this is the actual melee weapon. I’m more partial to the short sword for monster hunting due to it being easier to conceal than a long sword, it’s important to remember in accordance with werewolves that in general swords are not good for hunting animals. They’re meant for humans, not nine foot snarling death beasts. If you have the room then the traditional melee weapon for hunting animals of this kind is actually the spear. The spear’s length gives it substantially stopping power when it comes to the greater force of impact that an animal like a boar can generate when it charges or a wolf can when it lunges. The spear also allows you to keep the range advantage with a much larger enemy that you give up in choosing a shorter weapon. However, in a modern setting the spear will have a disadvantage indoors and give up a lot of maneuverability as an all-purpose weapon that your hunter may not be able to afford. Keep the knife or short sword as a means of taking off the head after the werewolf has been disabled and as a last ditch.
Use any variety of European longsword for the coolness factor or denotation of the One True Hero. The long sword itself is symbolic in Western Literature, so if you want your hero to be a Chosen One or a Noble Undercover then the sword is a great way to point to that without dropping a prophecy to accompany it. Just make sure to recognize the fantasy tropes and what they promise your audience because the use of tropes can mean making promises you’re unaware of and don’t intend to keep. The longsword more than any other medieval weapon has special/important cultural meaning in Western mythology/storytelling. If you’d like to avoid the standard fantasy tropes either subvert them or avoid the sword all together.
The katana is also highly symbolic if that’s your choice, however in the hands of anyone not Japanese it can easily become laden with Orientalism. So, be careful, or at least research the accompanying symbolism both for Western audiences in the recent decades and it’s meaning/importance to Japanese culture.
I hope that helps.