This little video shows the rules that govern the appearance of an Eligible Girl at the Ganesha Cup Boat Race to join your 'Team':
Although I made a statement to the effect that you would not be able to have a Girl on your Team until the 2nd Summer, it probably would be possible to ply Mist with Turnips on a daily basis from 2 Spring onward, sweep her off her feet for a Date on 19 Summer and have her offer to join your Team at the Boat Race in the 1st Summer...
Mist would be the easiest to target for this as Turnips are an inexpensive Spring Crop that grow quickly and you will meet her on the 1st day of the game. Actually, you probably could court Melody using Wild Coloured Grasses as gifts as quickly if you were determined.
Introduction Events for many of the Eligible Girls cannot occur before the middle of the 1st Spring, making it unlikely that you could advance your Courtship quickly enough to ask them for a Date before 19 Summer.