Well, to me in "the gift we give tomorrow", "tomorrow" answers to "when do we give it ?" not "to whom do we give it ?". But that may be because I'm not a native English speaker, in English the "to" is indeed not mandatory, so the sentence could technically mean both. In French the "to" (well "à") is mandatory, so my french-wired brain probably stops scanning once it found a way to interpret the sentence in a french-friendly way. Unless directly asked "could there be a second meaning ?" and then it'll dig further into non-frenchy ways. That's the way I feel it at least...

But IMHO, better use a sentence that can only be understood one way, when possible, especially when trying to convey a very strong meaning.