There is no straightforward rule for this so you'll have to learn to some extent. However, you can definitely get some hints in many cases provided you speak English (which has many common roots with French).
As a reminder, circumflex accents are used:
- to replace a "lost" letter (fenêtre - fenestron);
- to modify pronounciation (pôle, arôme, infâme, grâce);
- to distinguish between homonyms (mur - mûr / sur - sûr);
- in conjugation of verbs (chantâmes, as you mentioned).
Case 4 obeys grammar rules and is not part of your question.
Cases 2 and 3: I am afraid you have to learn by heart, but thoses cases are not so frequent.
Case 1: that is the most frequent case, and luckily that's also were you can use your knowledge about languages or etymology. You have to remember words from the same family, where one has a circumflex accent and the other has an additionnal letter (which usually is an "s"). So if you know the word "fenestron", you'll easily remember the circumflex accent in "fenêtre".
Here is a list of French words, followed by English (and some French) words where an additional "s" reminds you to put a circumflex accent in the French word.
Ancêtre - ancestor (French: ancestral)
Août - August
Apôtre - apostle
Bête - beast (French: bestial)
Château, châtelain - castle
Châtier - to chastise, to castigate
Conquête - conquest
Côte - coast
Coût, coûter - cost
Dépôt - deposit
Fête, fêter - feast (French: festoyer, festin, festivités, festival)
Forêt - forest (French: forestier, déforestation)
Goût, dégoûter - to disgust (French: gustatif)
Guêpe - wasp
Hâte, hâter - hast
Honnête, honnêteté - honest
Hôpital - hospital (French: hospitaliser)
Hôtel, hôte, hôtesse - hostel, host, guest
Île - island, isolation (French: isolé, isolation)
Intérêt - interest, interesting
Maître, maîtrise, maîtriser - master (French: bourgmestre)
Mât - mast
Pâques - paschal (French: pascal)
Pâte, pâté, pâtisserie - paste
Prêtre - priest
Quête - quest
Rôti, rôtir - to roast
Tâche - task
Tempête - tempest