Sorry I'm so darn late. It's our anniversary; 16 years ago my now OH and I began chatting about vintage sailboats across the Atlantic.
Pain after placing new fillings can have several causes, and even the best dentist cannot entirely aprevent them (although it's easy to raise the probability by sloppy work). The problem mentioned by Mélusine is an extremely rare thing, by the way. I haven't seen a single case in more than 40 years.
Clove oil (eugenol) works if it can reach the irritated nerve; that's why it's most effective when applied inside an unfilled cavity, where the dentinal tubules
are exposed. Those microscopic channels guide the oil to the nerve.
Dentinal tubules are also accessible from the gum line. The older you are the more tubules are exposed there. Yes, they also cause cercival hypersensitivity in those areas. Anyway, since your cavities got filled, the gum line is the point of attack.
Clove oil does have a fairly strong soothing effect on an irritated dental nerve provided it can reach it. It is used in many dental products. And yes NAT, you're right, it can irritate the gums, but when it comes in the form of a crushed clove, that's a minor problem. And chewing on aspirin can have a similar effect indeed, both soothing the nerve and irritating the gums.
Actually, clove oil is very concentrated stuff. You can also use it as a substitute for cloves in the kitchen, but it's so strong that you can replace five cloves by dunking the tip of a wooden toothpick into it by no more than 2 mm and then throw in the sauce or whatever (to be removed before serving! Kids, please do not try this at home!). The crushed cloves with a little bit of oil should be sufficient; if not, you may be a case for the regional dental emergency service.
Apart from topical application of clove oil and aspirin (or most other ordinary pain killers), Single Malt Scotch may help too, preferably when used not only topically but also systemically. I'm fairly sure some other, more profane booze will do the job too, I just never tried, but my baby brother says that my own moonshine plonk, which is extremely strong, works too. However, don't let yourself get caught by the breathalizer brigade after a systemic dosis.