usage - Using "a tiny" in the same way as "a little"

Saying

That made me a little happier

is clearly perfectly fine, yet no one would really ever say

That made me a tiny happier,

even though both "little/tiny bit happier" are fine.

Is there some logic to this? It's perfectly clear what the second sentence means, even though it sounds "a tiny" wrong.

Answer

This is similar to what Matt Эллен said, but uses the analysis and terminology of the Cambridge Grammar of the English Language. First, tiny is only ever an adjective. As such it can function as attributive modifier of nouns (e.g. a tiny flower) or predicative complement of verbs (e.g., that flower is tiny), and a few other minor functions.

The situation with little is similar when it's an adjective, but it's not just an adjective. We also have the determinative little (e.g., little water) and the complex determinative a little (e.g., a little water; a complex determinative is a single determinative made up of two or more words.) Determinatives typically function as determiner of nouns (e.g., that water), but some determinatives also function as modifier of adjectives (e.g., it wasn't that big). Most adjectives, tiny included, can't function as modifier of adjectives. Also, some NPs can modify adjective (e.g., [a city block] longer).

That made me [a little] happier 

D a little is modifier of Adj happier

*That made me tiny happier

Adj tiny can't modify Adj happier

*That made me a tiny happier

D a can't form a constituent with Adj tiny or Adj happier; Adj tiny still can't modify Adj happier

a bit

D a is determiner of N bit forming an NP

That made me [a bit] happier

NP a bit is modifier of Adj happier

a little/tiny bit

D a is determiner of N bit; Adj little/tiny is modifier of N bit

That made me [a little/tiny bit] happier

NP a little/tiny bit is modifier of Adj happier

The interesting question is how little became a determinative and how a little became a complex determinative.