Perhaps, just get rid of districts. If someone from across my state represents me better than someone local, then perhaps my appointment should not be limited by borders drawn for an election system that would no longer be in place.
Why even restrict the choice of representative to someone in your state? I'd just let anyone interest in the job apply to serve as a representative, provided they could meet some minimum number of votes nation-wide—perhaps 0.1% of the eligible voting population, so there could be at most 1,000 representatives. In practice it would probably be much less than 1,000, with a few individuals representing the major factions but plenty of room for minority positions. Each eligible voter gets three votes, and thus up to three representatives, which they are given the opportunity to change at regular intervals (e.g. quarterly, or when one of their representatives steps down). The votes are persistent until changed, and can be concentrated or spread out according to the voter's preference. A representative's influence in the House is determined by how many votes he or she currently holds.
This would, of course, be separate from the states' representatives in the Senate, to be appointed by the state legislatures. Popular representation is all well and good, but someone has to look out for the long term. Under my system the House would be able to approve any short-term (discretionary) expenditures unilaterally out of existing savings, but a 2/3 super-majority in the Senate would be required for anything requiring new debt (to include any increase in the money supply), speculation on future revenues, or a commitment of more than a few years. Finally, all laws would be required to maintain the approval of a simple majority in both the House and the Senate or face immediate repeal following a call for a vote.