For example, in in lisp-mode, I want to be able to use d as normal but also remap something like "dh" to something like "sp-splice-sexp-killing-backward". In vim you can remap keys like this without a problem, but with evil this would get rid of d's functionality altogether. Would there be some way to wrap this so that if a mapping exists for a d sequence, that action is taken and otherwise "evil-delete" is called with the key(s) instead?
As you've recognized correctly,
evil isn't designed to support arbitrary mappings. The crux lies within the way Emacs keymaps work; to put it simply, if you press a key, it either resolves to a command or a prefix/keymap and waits for the next one. You cannot have both.
evil chose to go for the first option and reads in a key by itself to determine the operand.
What is possible though is replacing the command with a custom that dispatches to either executing your custom command or the regular
evil operator. Possible ways of doing so are elaborated upon in http://permalink.gmane.org/gmane.emacs.vim-emulation/2000.