In a dedicated Gintama mode, the best starting guesses are the characters that give you the clearest visual and release-year anchors inside the franchise. Because the series is already fixed, your opener should help separate the remaining cast by hair, eyes, and era as quickly as possible.
Gintama mode draws on 5 characters and every one of them is listed for 2006, so the year tile is dead weight on this board and the solve has to come from visual clues and process of elimination.
The pool splits 4 male to 1 female, so a gender match usually removes about 4 of the 5 candidates in one go.
White hair is the tightest visual filter here, belonging to only 1 of the 5, while black hair is the broadest group with 3.
On eyes, red is the rarest tag (1 character) and blue is the most common (2).
Kotaro Katsura and Shinpachi Shimura agree on 4 of the five clue traits, and that pair is what decides most failed runs in this mode.
If you only memorise one character first, make it Gintoki Sakata — it carries the most distinctive clue profile in the Gintama pool.
Your opener should probe the widest axis left in the pool, because the series clue is already spent. Gintama mode has 5 characters, all listed for 2006, where white hair appears on only 1 of them, while black covers 3.
Avoid opening on white hair: it belongs to only 1 character, so a miss there teaches you almost nothing. A better second guess changes era or eye colour, and Gintoki Sakata is a safe anchor for that job.