Red mite don't live on the hens. If you see parasites on the hen during the day its very unlikely to be red mite. Its possibly lice or if you are really unlucky it could be Great Northern Fowl Mite, Feather/De Pluming Mite or Scaley Leg Mite (if legs and feet unhealthy looking.
For any of them I'd Poultry Shield house thoroughly, probably use a Red Mite Fogger then paint walls and nooks and crannys with white vinegar solution and finish off with Diatomous Earth or mite powder. If you had a separate coop to put the bird in for a few weeks I'd also do that as unfed red mites die off after a week. Ones that have fed can live alot longer hence how I'd treat the coop above.
Red mite - dust the bird every few days with DE and really rub it in everywhere. It might take a few attempts - I would read up on the lifecycles.
Never had lice (I don't think anyway), but have had what I think is Great Northern Fowl Mite, they live on the bird, feed continuously and will eventually kill the hen if you can't get rid of them. The bird can also suffer from some pretty nasty side affects including paralysis (so it can appear like Mareks which is what I thought I was initially dealing with). I will probably get blasted on here, but having had a couple of hens suffer and die, once I realised what the problem really was I took drastic action to save my flock.
I read up on Great Northern Fowl Mite and there doesn't appear to be any proper treatments that has been tested for hens or poultry used for eggs or meat. Treatments for Red Mite or Lice do not work with Great Northern Fowl Mite. But there are things which are used with other birds such as ornamental fowl, pigeons, parrots and exotic birds which do. It works and has been scientifically tested, but has not been scientifically tested for poultry. Its called Invermectin (have a look on the FSF website). We do not eat our hens - just eggs, so on advice from Vets we composted the eggs for 3 weeks (Vet said 5 days, but I wanted to be sure - Vet will not prescribe this). If Hybrid size you put 3 drops between shoulders, a drop under each wing and couple of drops round vent.
It worked straight away, but you have to be vigilant because life cycle may mean eggs hatch and you have to do it for second time and recalculate egg withdrawal.
I'd try other approaches first for Red Mite and Lice, but if it doesn't work and they are on the bird I would consider if its Great Northern Fowl Mite and treat as above.
I don't like using chemicals and I prefer a more holistic and organic approach, but for worming I always now use Marriages Feed with Flubenvet (no egg withdrawal) and in the case of persistent mites - Invermectin with a suitable egg withdrawal.
OP can I just ask if you ever brought in new hens with existing bird/s before?? Its not usually very easy. If you've done it before then you know what I mean, if you haven't then hope you find the following helpful.
If you have a solitary hen she is very lonely, but you will have considerable difficulty introducing extra hens especially if you only have one. I have a friend who has only 1 hen left after the fox took the rest - she's very lonely but my friend is going to wait until she dies before getting more. This is because its a long and difficult process. Existing birds will often attack and can even kill new hens. I have successfully introduced new hens into my flock periodically, but you need to keep the new and old flock living next to each other but physically separate for weeks, sometimes months. I do it by using 2 coops, 2 feeders/drinkers and dividing the enclosure up, then gradually putting one of the more placid birds in with the new hens under supervision and over time adding more hens until I just integrate them fully.
There is always bullying and pecking. This has to happen to establish flock order, but there can be fights on occasion and bloodletting. I always supervise with a garden hose for standby to break up more nasty attacks. Stockholm Tar and Anti Pecking sprays helps if a particular hen is being targetted.
Good luck and best wishes.