I do think at this age it's less that they are actually hungry all the time, and more that they simply find the sensation of being put down to be utterly alien to them - it's so different to being in the womb. If you have a look at the stuff written about the fourth trimester a lot of it makes sense.
In the womb everything was:
Dark
Quiet/muffled
Restricted/enclosed
Consistent
Cradled
Warm
Constant stream of nutrition
Moving/swaying constantly from mum's body movement
White noise of your heartbeat, blood, digestion
Whereas being put down in a cot or Moses basket - it's flat, open, he can move his arms and legs around which startles him, he can feel air currents and the temperature may change, noises are much sharper and there is no constant, things change a lot, he's having new and unusual sensations in terms of digestion etc.
So when your DH has him and is trying to let you have a break he will have the most success if he can recreate as many of these conditions as possible.
Keep lighting low (or at least not overly bright)
Noise should be without spikes. He can talk, babies often enjoy dads' voices because they recognise them from being in the womb. He could also use white noise to help. The best app we've found for this is a free one called Baby Sleep with an icon of a purple swaddled baby. Also making the "Sssssh" sound reminds babies of womb sounds which is why we instinctively do it to soothe.
Swaddle the baby or cuddle him in close so that he feels enclosed and wrapped up. They often prefer to have their legs bunched up than stretched out at this age.
Hold the baby against his own body/in his arms or try placing him in something curved, not flat, like a bouncy chair, or on one of those Sleepyhead things (note this is not recommended for night time sleep, but is OK if DH is awake and supervising)
Protect the baby from air currents although they do recommend not to let them get too warm because of SIDS - it's one you have to find the balance for yourself. Again hold him against his own body heat or try pre-warming surfaces before he puts him down. Don't put any direct heat source like a hot water bottle in with the baby as it can burn them, even if it only feels warm to you.
Movement - either rocking or swaying or jiggling himself, or using things like a baby swing, bouncy chair, rocker, or going out and about - car seat, pram/pushchair are all soothing for tiny ones.
Avoid unnecessary discomfort - change nappy regularly, keep baby in nice soft clothes without too many seams or elastic waistbands, check fingers and toes for stray hairs or threads wrapped around them.
With my babies I always offered them the end of my (clean) little finger to suck and if they seemed to find that comforting then I'd offer a dummy, both went off it by 6 months which was helpful as we didn't have to wean it later.