Smith quit the post on Monday amid a power struggle, which sees a group of shareholders aim to remove key figures from the board, including chief executive Craig Mather.
In a statement released to Press Association Sport, Smith called for fans to get behind formal moves to install former newco director Paul Murray and Scottish accountant Frank Blin to the board.
Asked for his reaction to Smith's departure, following a 1-1 draw with Newcastle in a friendly, McCoist said: "I'm obviously very disappointed - that goes without saying. I'm very disappointed indeed."
Asked if he had concerns that Smith's exit leaves himself isolated, the Light Blues boss added: "I don't really think so, to be honest with you. I don't think there is an awful lot more Walter could have done.
"He has got his own reasons for his departure. I respect that, totally respect it. Once again, there is no doubt, it will be with the club's best interests at heart, whatever has forced him to make that decision."
The visit of Newcastle witnessed the return of former chief executive Charles Green to Ibrox in his new role as club consultant.
The Yorkshireman and McCoist were involved in an angry exchange of words over the weekend after Green claimed in a newspaper article that the manager must deliver a cup, as well as the league title, this season.
McCoist responded by branding Green "an embarrassment" in a blistering post-match interview after Rangers crashed out of the Scottish League Cup to Forfar.
Both men sat several rows apart in the directors' box on Tuesday night, before McCoist took his usual spot in the dugout for the second half. The manager confirmed the pair did not speak but refused to be drawn any further.
McCoist said: "I'll tell you how much it's not turning into a war of words, I'll talk to you about football tonight. I'm only interested in the game to be honest with you and I'm thrilled with the game, absolutely delighted with it.
"From start to finish, against a top team, I thought we played really well. Saturday was a massive disappointment, to go out of the cup. The reaction from the players and, indeed, the supporters towards the players was fantastic."
McCoist sat with Mather during the first half and said there were no issues between himself and the chief executive.
He added: "He's the chief executive of the club so I'm in regular discussions with Craig. We speak regularly and we meet regularly.
"He's the chief executive of the football club and as I've said all along, I think the chief executive and the manager is arguably the biggest working relationship at the club. It's safe to say, and see, that we are absolutely getting on fine."