Chaotic melting points of n-alkyl carboxylic acids
My guess without looking at any literature:

Higher acidity $\implies$ higher hydrogen bonding $\implies$ higher melting point.

Thus, changing from H and Me where there is virtually no +I effect from the chain, to Et, Pr, Bu where the +I effect of the chain decreases the melting point, as the hydrogen bonds get weaker, as the COOH groups are less acidic.

Further increasing the chain length after hexyl does not really lead to any additional +I effects on the COOH group that is now far away. Thus, the melting point increases as there are additional van der Waals interactions between the longer chains.

Then there seems to be a factor that makes crystals of odd chains more favorable.