I'd like to see something like an All-Around competition like they do in gymnastics, except I don't think there's time for another separate event, so I came up with a cool concept.
I definitely think a team element could be included, but run as part of the singles and doubles competition. Countries that want to be involved in the "All-Around" competition have to field competitors in both singles and doubles events to qualify. For example, Justine is the only competitor for Belgium in singles. Since there's not Belgian doubles team, Belgium does not qualify for the women's "All-Around" event.
So, depending on how far players from each country progresses, their points are added to the teams total. Because some countries have more players playing than others, the points could be added and then divided by the number of players to get an average. So, if for example, Maria Vento-Kabchi is the only competitor for Venezuela in singles, her points go onto the tally for Venezuela. If Milagros Sequera was also playing in singles, you would add both their points together and then divide by two to get an average. This would be the amount of points that would go on their tally. It's the same for doubles. If there are two teams, their points earned are divided by two to get their total and then added to their tally.
For example: Myskina earns 365 pts, Dementieva 1pt, Petrova 132 pts, Kuznetsova 200 pts, their average is 174.5. In doubles, Kuznetsova and Elena Likhovtseva earn 340 pts, while Myskina and Dementieva earn 23 pts, their average is 181.5. Russia's total score is 356. Do this for all the teams eligible for the "All-Around" competition.
At the conclusion of the competition, the nation with the most number of points won, wins the "All-Around" competition and receives the teams Gold, second receives Silver and third Bronze.