Here's a list of all head coaches with 10 or more playoff wins, and their records in years 1-3. I am encouraged by what I found. If you pick the right coach, you may not need wait long for the turnaround.
Belichick: 11 playoff wins, 20-28 in his first 3 seasons in CLE, 11-5 in 2nd year at NE
Cowher: 10 playoff wins,, 32-16 in his first 3 seasons in PGH, 11-5 his first year
Gibbs: 17 playoff wins, 30-11 in his first 3 seasons in PGH, 8-1 his 2nd year there
Grant: 10 playoff wins, 23-16 in his first 3 seasons in MIN, 12-2 his 3rd & 4th year
Holmgren: 10 playoff wins, 27-21 in his first 3 seasons in GB, 9-7 his 1st year in SEA
Levy: 11 playoff wins, 19-29 in his first 3 seasons in KC, 12-4 his 3rd year in BUF
Lombardi: 10 playoff wins, 15-9 in his first 2 seasons in GB, 35-6 in the next 3
Noll: 16 playoff wins, 27-21 in his first 3 seasons in PGH, 11-3 his 3rd year there
Parsells: 11 playoff wins, 22-25 in his first 3 seasons in NYG, 10-6 his 2nd year in NE, 12-4 his 2nd year in NYJ, 10-6 his 1st year in DAL
Reeves: 11 playoff wins, 27-21 in his first 3 seasons in GB, 10-6 his 1st year in DEN, 11-5 his first year in NYG, 14-2 his 2nd year in ATL
Shula: 19 playoff wins, 30-11 in his first 3 seasons in BAL, 10-4 his 1st year in MIA, 60-17 in his first 6 years in MIA
Walsh: 10 playoff wins, 8-24 in his first 2 seasons in SF, 13-3 his 3rd year
The notable exceptions (where a team stuck with them):
Landry: 10 playoff wins, 9-28 in his first 3 seasons in DAL, first winning record came in his 7th year
Seifert: 10 playoff wins, 38-10 in his first 3 seasons in SF, a dismal 16-32 in CAR