As nothing has been made official yet you aren't committed to anything so you do still have some options, albeit limited ones.
I'm assuming here that the "too low" remark refers to your presented wage compared with the market rather than it being too low for your own financial situation.
Option 1: Accept the current figure and play the long game
Yu can take the offer as it stands and continue on in the job and and try and make the best of it. Perform the job to the best of your abilities and try and use that and your reseach of the market to try and correct the figure at the next negotiation opportunity. It does mean you will likely be behind the curve for a while in terms of wages but at least your job search will be done for now.
Option 2: Try and re-open negotiations
You can try and go back to them and say that you have reviewed the number you previously stated and that you consider it too low. This is unlikely to be well recieved, even assuming the company would be willing to pay the higher number (not gauranteed given their intitial offer was even lower then your counter) then at best you look a bit indecisive and at worst you look a bit underhanded for changing the goalposts on them and this may well affect your working experience with this company. If you aren't planning on this being a long term job though then it might work out ok for you. Of course there is the possibility that they will either stick at the current offer (in which case you may have harmed your reputation with them for no gain) or they may just pull the offer and your job hunt goes back to square one.
Option 3: Act as if you are going to walk away
You could give them a polite but generic refusal to the offer, and if they are keen to secure your services then they may ask if there is anything they can do to change your mind. Then you can come back with a higher figure. It's a bit sneaky and you really need to be comfortable with the possibility that they will just let you walk but it's probably the best way of getting more money right now from them without harming your reputation in their eyes.