The Norwegian's transfer plans looked to be in tatters after his side were dumped out of Europe's premier club competition by Legia Warsaw this week, the Poles winning 2-0 at Murrayfield on Wednesday to claim a 6-1 aggregate triumph.
But the Hoops could be handed a shock second chance after Uefa launched a probe into allegations that Legia fielded an ineligible player, defender Bartosz Bereszynski, in the final four minutes of the second leg.
If found guilty, the Poles could be kicked out of the competition and Celtic given another shot at reaching the money-spinning group stages.
Hay told SportTimes: "Even if Celtic do get back into the Champions League, that doesn't take away the fact that they were poor over two legs against Legia and Ronny still has a lot of work to do to get the team playing a lot better than they did.
"If they were to get back in, the players there would have to improve their level of performance and I think one or two acquisitions would have to be made as well.
"After what has happened, if Celtic did get back in, I think they would push to get a couple of players in prior to the play-off round.
"If they say they have targets they are thinking about getting, why not get them in now?
"That is if they are lucky enough to get back in the Champions League, of course."
Boss Deila added just two players - Craig Gordon and Jo Inge Berget - prior to the Legia clash as the Hoops board kept their powder dry in the transfer market.
The strategy provoked a furious reaction from fans in the aftermath of Wednesday's defeat and Hay has urged the Parkhead powerbrokers to loosen the purse strings to boost Celtic's chances of mixing it with Europe's elite.
He said: "I don't think it is necessarily about the transfer fee. I think Celtic have to consider breaking their wage structure.
"I know it is difficult in Scotland but, for the sake of making it easier to get through these qualifiers, you have to get a better quality of player.
"You have to invest at a level where you are able to attract better players.
"I think the board have to consider upping the ante for the basic wage. That has to be done.
"To entice a player to get you into the Champions League, you have to increase the salary on offer.
"I don't want it to be a damaging amount, it has to be affordable, but they need something to appease the fans.
"We are only talking about a couple of positions - striker and creative midfielder in particular - but that bit of quality can be the difference between making the Champions League or not."