Are there any tools to reconstruct alternative haplotypes to the genome reference from short read sequencing data? For example, if one has prior knowledge that there might be duplicated or diverged haplotypes for gene X in the resequencing of a human individual, what tools can be used to reconstruct the putative alternative haplotypes in that gene X?
It sounds like you are looking for haplotype phasing. GATK has a short read phasing walker that works given an input VCF file of variants and the alignment reads in BAM format:
There are also more inference based methods. Wikipedia has a good treatment with links to several other packages: