OP: I have done a very similar job and I think YANBU for worrying, but IMO it would be best to let it be.
If he really can't bear to lower himself to even THINKING ABOUT going to a food bank for £30-40 quid worth of food - things can't be that bad. If he was previously better off, he more than likely has a small cushion to live off or friends/family to support him. In this scenario I doubt a gesture from you would be required. Cynical I know but people really do find ways to get by all the time, he is not unique. If you are that desperate then pride will not get in the way of a good meal. You have offered him the options to keep himself going till this is sorted, food bank, sally army, etc, and it is up to him to choose to accept them or not.
As an aside, he can reclaim housing benefit straight away if he has no other money coming in. So he can either live off that or use it to not go into arrears with the rent. This applies to anyone whose out of work benefit has been stopped be it jobseeker's, income support, ESA etc, if they have no other major income (child tax credits doesn't count I don't think), they can go straight to the council and reapply for backdated Housing Benefit from date of stoppage due to change of circumstances with a bank statement/mini printout to prove no income, as it is solely based on the money you have coming in.
From what you put, he has lost his entitlement to jobseeker's allowance through no fault of his own and needs to appeal the sanction, appeal the refusal of hardship payment and get a crisis loan in the meantime. He is entitled to a loan no matter what he has said to you unless he has borrowed hugely before, and I really doubt the emergency hotline would turn down someone with 0 income! Even then you call back, get someone different and try again till you get the money. Just like when your delivery doesn't arrive or your shopping goes manky before the BBE.
You have a choice, you can have your heart broken every day or you can assume that the people calling you are adults who, for the most part, can manage their own affairs and seek out help when they need it rather than looking to a complete stranger to rescue them. By and large people are resourceful and can, and will, survive rough patches like this man is going through. I have sympathy, but I cannot be everyone's mum and go in guns blazing for everyone I talk to. You are there to equip them with information to do that themselves, and that is all you CAN do.
I think that for your own sake you need to take a step back from the people you talk to on the phone and not get too involved emotionally, because it DOES start to affect you psychologically after a while, and you do get very stressed, especially when you never meet the people you talk to and have no follow up or closure. You simply can not be a bottomless well of sympathy all day every day without it affecting you. 
I would be cautious about contacting him privately if I were you - not sure what you do or where you work but data protection law would worry me. Imagine also if his neighbours started asking why he got a food parcel / Ocado delivery and who the note was from (you don't know if he'll even be in, after all) - if he's too ashamed to go to a food bank then how would that make him feel?
If you really need to assuage your worry and guilt and you happen to work for the DWP, get involved, campaign politically, write to the man's MP citing an 'anonymous' caller, and leak the story (without any personal details) to the papers. The Mirror I believe ran an ex-ATOS nurse's expose of how badly ESA claimants were treated by ATOS staff and it did get Parliament attention.
I am probably going to have the entire thread come down on me like a ton of bricks now. Oh well. If it helps, I am living off rice and reduced veg at the moment as I am waiting for my DLA claim to go through. 11 weeks and counting...!