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:

1. to replace a "lost" letter (fenêtre – fenestron);
2. to modify pronounciation (pôle, arôme, infâme, grâce);
3. to distinguish between homonyms (mur – mûr / sur – sûr);
4. 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

Under the French spelling reforms of 1990, there are only four places one should use the circumflex on i and u:

1. The conjugation of the first and second person plural of the passé simple and the third person singular of the passé du subjonctif, e.g. nous vécûmes;
2. Where it indicates different pronunciation, e.g. jeune /ʒœn/ vs. jeûne /ʒøn/;
3. The masculine singular of the adjectives dû, mûr, and sûr, to distinguish them from du, mur, and sur; and
4. The forms of the verb croitre that, without a circumflex, could be mistaken for parallel forms of the verb croire (e.g. je crus [from croire] but je crûs [from croitre]).