It's not easy, but I'm inclined to think that Rangers fans should actually be grateful to Mark Warburton and the rest of the outgoing management team.
The reason for this is financial: if Rangers financial position been better, I suspect they would have sacked the manager some time ago - certainly after the recent defeat to Hearts, but probably after the previous defeat to Hearts. As it is, with the club existing on soft loans, I suspect they were resigned to keeping him in-post until the end of the season, as they simply couldn't afford the compensation they'd have to pay out. But that's not a case of wanting him as manager; that's rather a recognition that they couldn't not have him as manager.
As it is, by resigning in the manner he has, Mark Warburton has allowed Rangers to get a new manager in, and saved them that compensation that they couldn't afford. (Of course, the way that it has been done has been a shambles, but that's just par for the course for Rangers right now.)
Of course, the question is: what now?
Personally, I'm inclined to think Rangers should seek out the cheapest possible option for an interim manager to take them to the end of the season. The reality is that this season is a write-off, and has been for some months. They were never really in a title race, and they're not going to win the Scottish Cup (Celtic are basically unstoppable this year). All that matters now is getting into Europe - Rangers need to come at least third in the league, and even fourth might do it. Beyond that, it's just a matter of pride, but there are bigger issues.
Once they get to the end of the season, Rangers should take stock of where they are. There are two possibilities: either they have significant additional investment in the next few months, or they have to carry on much as they are. (And since they don't have that "significant investment" now, they can't really make the appropriate determinations now.)
If they have significant investment in place, Rangers can think of trying to close the gap to Celtic, in which case they want a top-quality manager, they want to back him with big funds, and go for it. (And by "big funds", I mean enough to bring in, effectively, a whole new team.)
But if, as seems more likely, they don't have any significant investment in place, then Rangers (and especially their fans) probably need to accept the hard truth that they can't stop Celtic getting ten, fifteen, or indeed however many in a row as they want - Rangers will effectively never catch Celtic unless and until Celtic implode. And to attempt to do so would be to make unwise spending that would probably kill the club - Rangers have just about come back from liquidation, but they won't do it again.
And in that case, the question is not "how do we close this gap?", it's rather "how do we get this club back onto a solid footing?" That is, sort out the finances first, so that surviving the season ceases to be a matter of soft loans, and the threats to the business are rather less existential.
And that means putting in a team that are going to build over a long time, with very solid scouting and development teams so they can, hopefully, replicate one of Celtic's great successes in recent year - their ability to find and develop young talent, get them cheap, sell them dear, and reap the rewards.