We don't know any phonetic details about Latin stress/accentuation.
The phonology of Latin stress is also somewhat uncertain, but it is not thought that there was a phonologically distinctive contrast in Latin between different types of stress contours for a syllable.
("Coepio" might be a confusing example since apparently there is evidence that the "o" and "e" in this word were sometimes were pronounced as separate syllables. I would guess pronunciation as separate syllables might be theoretically possible wherever "oe" is derived from separate morphemes, rather than from an Old Latin diphthong. An example of "oe" that Lewis and Short seems to indicate was never pronounced as a diphthong is "coemo"—although there was a contracted form "como". Another similarly ambiguous sequence of letters is "eu" as in "neuter" < "ne + uter".)
Some ancient grammarians alleged there was a difference between "acute" and "circumflex" accentuation that was conditioned by the word's phonological form
Some Latin authors describe Latin syllables as being able to be pronounced with either an "acute" or "circumflex" accent, as in Greek, but there doesn't actually seem to have been a contrast in any context between acute and circumflex accentuation in Latin the way that there could be in Greek ultima syllables with long vowels. Rather, the described distribution of acute and circumflex accents in Latin is apparently predictable based on the vowel length and the position of the accent.
Sturtevent 1920 (p. 216) cites Donatus iv, p. 371 8 ff. K. as giving the following rules. The acute accent is used in most contexts; the circumflex accent was only supposed to be used when the accented vowel was long and occurred in a monosyllable or in the penult syllable of a word with a short vowel in the ultima.
Based on these rules, dīcō and coepiō would both have an acute accent, while pīla and rēs would have a circumflex accent.
But modern scholars tend to dismiss these accounts as misrepresentations based on the model of Greek accentuation.