EDITED: To answer your question, don't inform the new company yet.
Behind your question, I see many questions not specifically asked but I suggest your question could not be asked unless other important questions were there as a necessary root cause behind the one eventually asked "out loud" here. Indeed your final comment clearly reveals the presence of those unasked questions and just how fundamentally important they are to you.
Bargaining between businesses in the specific circumstances can be very risky. I can of course only guess at the unrevealed details, but with knowledge to hand of your situation, I would suggest finding out a few things first, then the answer to your question will almost present itself (but I will still give an answer):
Very quickly (yet very accurately) assesss the work situation where you are now. Do you work well with your peers? Is there mutual respect in evidence? Has management indicated before now that they value your expertise? How is their business acumen (in terms of bottom line, not what the company produces)? How long has the company been in existence?
These above are some of your urgent considerations. Once you have assessed them, you will have a part of the information you need to stay or go.
Now to focus on your present company:
How much of the market do they command, and for how long? Do they have a policy of monitoring their customer needs and responding to them? Would you be proud and happy to visit a client company and introduce yourself as working where you are now (and why - or why not?)? How does HR respond to social inter-action issues, and more importantly do you think the way HR does things is always ethical (not legal, take that as read, I mean ethical)?
And the 40% they offered you:
Why did they offer it? The obvious answer may not be the right one.
As a business, I would offer the least money if it was acceptable to to a competent employee and my shareholders would say that's a very good thing to do.
But after six months of seeing the quality of your work, even if they had not known you were about to resign, six months is still a highly understood (even if not spoken of) period of assessment for a new employee. You may have about to have been given a raise of some amount even so. The knowledge to management of what another prospective employer was prepared to offer you, has simply set the amount of that raise you didn't know was coming.
Finally, what are market conditions for jobseekers overall right now? Obviously six months ago when you took that low offer, conditions were bad for jobseekers. I would not automatically assume that it is easier to find a job now. There is an understanding extant, based on what evidence I don't know, that it is easier to find a job if you are presently employed when you seek, than if you are unemployed when you seek it. So what are ALL of the reasons, subtle and hard, why your prospective new employer offered you the job instead of someone else?
Unless your research on your new company comes up with glowing reports based on hard evidence, I would stay where you are if the company you are with now comes up looking good after you have done the assessments I have suggested.
If all of the above means you decide to stay, that is very good for you for other reasons. I may add these in an edit, but this is long enough.