Bilic was appointed as the Hammers' new boss on a three-year deal on Tuesday, returning to Upton Park following an 18-month playing stint under then-manager Redknapp 19 years ago.
He replaces Allardyce, who endured a tempestuous relationship with much of the West Ham support despite guiding them back to the Premier League and then retaining top-flight status for the next three years.
Co-chairmen David Gold and David Sullivan decided against offering Allardyce a new deal at the end of the season, with the 60-year-old departing just minutes after the final game of the campaign at Newcastle.
With the Hammers in the qualifying rounds for the Europa League, Redknapp feels Bilic has a great chance to build on an already well-established relationship with the fans.
"It is a big bonus for him," Redknapp told Press Association Sport.
"Sam did a great job but it seemed whatever he did was never good enough for a lot of the fans
Slaven comes in as a popular figure and it is a big advantage for him to come in knowing the fans will give him a great welcome when he arrives.
"The fans like him there, I think he will be a very popular new manager at West Ham
I was sad to see Sam go, he did a good job but they have got a decent group of players for Slaven to work with and the thought of going into a new stadium is good for them and I'm sure he will do well."
Redknapp praised the work already done elsewhere by Bilic, who he also revealed to be a 'real character'.
"He did a good job with the Croatian national team," Redknapp added.
"He has been around since then, went off to Russia and Turkey but now he has got a great job, managing in the Premier League at West Ham - a great club and a fantastic opportunity.
"He is a bit of a lad! He smokes about 80 fags a day, plays guitar in a pop-rock group and is a real character - but he really knows his football.
"It wouldn't surprise me if he brought in a few old faces back in there, one or two of the lads he played with
He might use them on the coaching side, it is just one of the feelings I have got."
One of those old faces is unlikely to be Rio Ferdinand after Bilic's former defensive partner retired last month.
The former England captain, who has also backed Bilic to succeed, has signed as a pundit with BT Sport - but Redknapp thinks Ferdinand will follow Bilic into a coaching career in the future.
He said: "Rio is doing his coaching badges and he wants to get into the game,
"He is no different to a lot of the lads who get into punditry but they can't all be as fortunate as Gary Neville who does the punditry and the England job.
"A lot of the lads just have to do the punditry but eventually he will want to get into the football world - in a few years time he will have done what he is doing with BT, watching a lot of football, but one day I'm sure he will get into management."
Source : PA
Source: PA