Disabling "slice_columns" does the trick:

my $te = HTML::TableExtract->new(
    keep_html=>1,
    headers =>[qw(RefSNP)],
    slice_columns=> 0);
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From the doc:
Enabled by default, this option controls whether vertical slices are returned from under headers that match. When disabled, all columns of the matching table are retained, regardles of whether they had a matching header above them. Disabling this also disables automap.

New output:

Table(1,0:
<strong>Organism:</strong>
human (<a href="http://www.ncbi.nlm.nih.gov/Taxonomy/Browser/wwwtax.cg
+i?mode=Info&id=9606"><em>Homo sapiens</em></a>)
[download]