A complete description of the formation of comparatives for adverbs is given in CoGEl (A comprehensive grammar of the English laguage). (It is complete as far as it treats the main points) The rule for adverbs is derived from that for adjectives.
Adjectives
(CoGEL § 7.81) Choice between inflectional and periphrastic comparison
The choice between inflectional and periphrastic comparison is largely
determined by the length of the adjective.
(a) Monosyllabic adjectives normally form their comparison by inflection:
low - lower - lowest
Real, right, wrong, and the preposition like take only periphrastic forms:
- She is more like her grandmother.
She is liker her grandmother.
However, most other monosyllabic adjectives can take either inflectional or periphrastic comparison.
(b) Many disyllabic adjectives can also take inflections, though they have the alternative of the periphrastic forms:
- Her childre are politer/more polite.
- Her childre are (the) politest/(the) most polite.
Disyllabic adjectives that can most readily take inflected forms are those ending in an unstressed vowel, [syllabic l, as in "able"] or [schwa (r coloured schwa in AmE)], eg:
-- y: early, easy, funny, happy, noisy, wealthy, pretty
-- ow: mellow, narrow, shallow
-- le: able, feeble, gentle, noble, simple
-- er, ure: clever, mature, obscure [user LPH's addition: not"eager","proper",etc.]
Aderbs
(CoGEL § 7.83) Comparison of adverbs
(e) Adverbs that are identical in form with adjectives take
inflections: fast, hard, late, long, quick. They follow the same spelling and phonological rules as for adjectives, eg:
early - earlier - earliest
When earlier is synonymous with before (that) or previously, and when later is synonymous with after (that), they are not comparatives of the adverbs early and late.
Note that "most
readily" in "7.81 b)" implies no unique choice and nothing as to the preference between the two use, which each user of English has to determine through experience; the following ngrams show that.


"More early" is not exactly a form that you could call wrong, but it is hardly used in comparison to "earlier".

Although the pages of examples for "more early" contain numerous occurrences, a patient examination will show that a great many of those are false positives (for instance, "more" does not modify the adjective but instead the noun it qualifies). Nevertheless, the periphrastic form is found often enough (even fairly recently) and it is sometimes used to some advantage, so that it seems possible to say that it is not really incorrect.
(ref., 1972)... And if this was the case then , we may safely conclude , that the Unitarians were much more numerous in a more early period , as it is well known that they kept losing , and not gaining ground , for several centuries
(ref., 2003) This evening Mr Hume not quite so well , and goes to bed at a more early hour than he used to do . Ferry bridge , Sunday , 28th
(ref., 1974) .. the latter more early useful ... than they should otherwise be
(ref., 1973) I would therefore recommend to every parent to begin the establishment of authority much more early than is commonly supposed to be possible
(ref., 2000) Open bite could even be correlated with yet inadequate growth of the jaws to accommodate the disproportionately large size of the more early - maturing tongue that must posture forward for accommodation within the oral cavity
(ref., 1807) Sometimes it happens , by badness of weather , & c . that a considerable quantity is fown the first week in december , but the more early fowings generally produce the strongest crops
(ref., 1858) The improvement is most strongly marked in the acute cases , and the more early the operation the more successful is the result obtained
(ref., 1858) but there are causes which operate in promoting its more early maturity
(ref.,1958) Pyrus serotina pear require more amount of nitrogen through the growing year , more moisturous and fertile soil , to start more early and rapid growth than apple eventhough the Korean weather condition in spring season is always
In this preceding case the use of the periphrastic form is quite understandable as "more" applies also to "rapid".
(ref., 1814) The sweetness of climate ad fertility of soil in Asia Minor, were favourable to a early icrease of population, and consequently to the more early organization of civil society.
This new occurrence seems justified by the need to make it clear that what is meant is not "preceding organization".