This "double ee" you mention, is that typed with one keystroke?
And what about the "ie" ?
Dead keys are a concept you can best get familiar with by experimenting
on a scratch layout while going through section 6.7 in the manual, and
I mean step by step.
Dead keys are called like that because they don’t produce any screen output by themselves: they delay the output by one keystroke.
Taking your situation as an example:
If h is a dead key character, pressing it activates the "dead key state" that is associated with it.
Then the key pressed after it can do two things:
1. if that key is defined as a normal key, it’ll produce output;
2. if that key is a dead key character too, it’ll activate another "dead key state", and you’ll have to type a third key before output is shown.
Ukelele can convert normal keys into dead keys and vice versa, and lets you create and edit dead key states too, so in principle all is possible, even if those "double ee" and "ie" are not typed with one keystroke currently. But it’ll make things much more complicated.
Could you attach the keyboard layout you’re working on? I’m rather in the dark as to your situation, and my whole questioning may be wrong.
In fact, it’ll probably be much easier for me to quickly make the desired change than trying to talk you out of a situation I don’t understand.