Right after the season ends, give an 8 day window until the official playoffs start. That lets teams rest up and develop strategy for their first round opponent.

During this 8 day period, all 14 teams that missed the playoffs play a single elimination tournament for the 1st pick (worst record remaining after 1st round gets second round bye). Every pick after that will be determined by point differential and record (worse you lose by, higher your draft chances but your record is assigned a multiplier based on where you landed...lose big and have the worse record, you'd get #2).

The seeding would be so that the worst teams played each other and the better teams played each other. So worst record vs second worst, third worst vs 4th worst, and so on.

This allows the worst teams better chances to get to the top and the better teams harder routes.

So for 2017 it would have been:

Brooklyn vs Phoenix
Second round bye so winner moves to final 4.

LA vs Philly
Orlando vs NY
Winners play each other

Minny vs Sac
Dallas vs NO
Winners play each other

CHA vs Det
Denver vs Miami
Ditto

These extra 3-4 games are played at one location, every other day, in a playoff city so no particular home court advantage but closest to the worst team if plausible (given some places will just be closer to some).

Each player will receive a bonus for playing in the extra games (15-25% of their annual salary, prorated based on advancing, winning gets you the full amount, lose early get an evenly prorated amount).

May the best loser win.