A user visits a proxy site like hidemyass while using Tor.
What would the implications be of using a proxy within Tor? Would that in theory enhance anonymity by adding another layer of security?
This depends a bit on how the two are configured.
If Tor is the outer connection, then the proxy would prevent the traffic from coming from an exit node; since the list of those is public, this could potentially get around restrictions from a site operator who wishes to block Tor traffic. It could also potentially serve as a layer of protection against a malicious exit node, as there's no encryption in the Tor protocol at that point; normally TLS (HTTPS) is used for that role, to prevent eavesdropping (e.g. a session stealing attack) or modification of the response (e.g. adding in advertisements or malware).
A free proxy site, however, is likely to be less trustworthy than Tor, though, which makes these benefits negligible.
If Tor is the inner connection, the proxy could be used to bypass restrictions by the ISP aimed at stopping you from connecting to the Tor network. However, a common proxy like hidemyass is probably also blocked, and bridges are a more direct way of dealing with that problem.
Overall, I think the primary result would be a slower connection, with minor to negative impacts on privacy and security.