1. There is a broad correlation between brain size and intelligence - as a brain must be of a certain size to have a certain complexity.
But the correlation is crude. Some big animals need big brains to control their bodies - some relatively small animals seem relatively very intelligent - I am thinking of some parrots.
This is because the brain is a multi-functional organ - most of it is concerned with non-'g' matters; and intelligence differences are probably only underpinned by a part, probably a small part, of this organ.
2. My understanding is that intelligence is mostly about efficiency - and efficiency specifically, flexible and multi-functional efficiency) requires complexity; and complexity of the brain is constrained by size and also developmental speed - on the whole a more efficient brain will tend to be larger and take longer to develop (construct) than a less efficient brain.
(I do not think it would be biologically possible for an organism to develop and mature an extra intelligent (therefore more complex) brain and also to do it more rapidly than usual - therefore high intelligence pretty much requires a longer, and either slower or not-faster, period of brain development and maturation during childhood. Therefore, measuring brain size during childhood is not likely to be a good guide to intelligence - a small and simpler brain may develop and mature quicker than what will eventually turn-out to be a larger and more complex and more efficient brain.)
However, part of making a more efficient brain is probably developing a denser connectivity of brain circuits, rather than simply larger numbers of circuits. The 'g' advantage of men over women is (I would guess) probably related to greater density of connectivity, more than to a physically (and proportionately) larger brain.
3. Brains can be made smaller, and intelligence reduced, by a range of pathologies - illnesses, genetic and chromosomal problems. This is trivially true. But it means that in correlation studies there will be some small brained people with low IQ simply due to damage and destruction of various types (and there is an unconstrainedly large number of causes of brain and intelligence damage).
4. What about larger than average brains? Well, brain size might also be increased by some pathologies - but there may be other, non-g causes for brains to evolve or develop extra--large specialized circuits, to serve specialized functions - perhaps in vision or memory (eg. the suggestion that some specific brain regions are expanded in Australian Aborigines to enhance specific visual memory functions).
But IF (and it is a big IF) these specialized regions could be controlled for and excluded from the analysis - and analysis focused only on 'g' relevant brain regions (currently unknown); and if the brain substance was as densely connected as normal - then increased intelligence would be constrained by brain size: in other words, I think the only plausible way that intelligence could be increased by evolutionary pressures would be to increase the complexity of connectivity, which would probably entail an increase in the size of the (currently unknown) relevant parts of the brain.
5. I have been talking about brain size - head size is only loosely correlated to brain size, and subject to further pathologies. And hat size is only loosely correlated with head size - the shape of the head can make a difference here (so big headed people may only fit a small hat!). I don't think that measures of head or hat size can contribute much or at all to unravelling the difficult aspects of intelligence - except in a negative way:
Shrinking heads (and hats) over time (declining adult head circumference) would probably be strong evidence of declining intelligence - but the opposite would not be true for the reasons outlined above.