He sounds just as conflicted as so many stay-at-home moms - "need to feel like a man" means earning money (?) but staying at home is frustrating because the work just piles up. Housekeeping and child care can get "old", and even with "just one" the other two still come home from school and contribute to the mess. They also have homework and activities and play dates, and there's a lot of running around just when the youngest is getting tired and cranky. How about all the men who criticize women who stay home "and do nothing" all day while the kids are at school?
You and your husband are going through what many other couples go through, but with the roles reversed. The working spouse doesn't feel the house is kept up to appropriate standards and that the work isn't all that hard, and the at-home spouse is frustrated because it's not always so stimulating or it's "getting old" or it's not fulfilling or there's too much isolation, or a zillion other things. It may be harder for your SAHD because fewer employers are less sympathetic to or interested in men who have been out of the job market for a while. And jobs can be tough to some by in a lot of areas anyway.
Doing a temp-to-perm job might be great, if you can manage the extra expense of day care - that would have to be arranged on a daily basis even though there might be days when there was no actual work available, so you'd have to budget for the added expense up front. You'd probably have to have someone in the house for the baby as well as for when the older kids get off the bus.
Getting "any" job isn't always something that makes the returning-to-work spouse feel particularly valuable or contributing, and it's worse if he/she is at the stage of feeling that the stay-at-home job wasn't done so well. He could be depressed - and one of the key symptoms of this is inertia, not being able to decide what to do next. Task completion is very tough for a person with depression, and so is sorting through available job postings.
So to start, maybe the two of you can cut back your expectations of how "done" the house has to be. Maybe the 2 older kids can take on a little more responsibility - maybe put 3 baskets or bins in the public rooms (family room, living room, whatever) for all the kid stuff - toys, stray gloves, etc. Everybody takes 5 minutes to do a "sweep" of the rooms and everyone's stuff gets put in the right basket. One kid can set the table, another can dust the end tables. Kids can sort their own socks and maybe fold their own laundry. Everybody cooks together on Sunday afternoon to make at least 3 meals for the coming week.
Maybe it's worth investing some money with a career or life coach, or someone who does resume preparation for someone returning to work. I have a friend who does resumes, for example, for about $350 - that doesn't mean just typing it, it means really digging to put in the most important info that employers want to see. For another $150 or so, she adds in services like writing a cover letter that is focused and in tune with what employers are looking for. Resumes today are scanned by HR people at a ridiculous rate - something like 12 seconds. So there are resume prep people who have a very high success rate, which is defined as getting you the interview. There must be someone in your area, although these coaches work across the phone and the internet as well so you might find someone he likes who is at a distance.
My husband has worked at home for many years and he was a hands-on dad. He was laid off when I was in labor with our son, so we had little warning. His company went out of business shortly thereafter, and the whole industry was hurting, so there were no jobs. I was in no condition to be helpful or even emotionally supportive. It's not the same as what you're going through but we had our stresses. We each went through a time of high expectations and of wanting the other one to do more. So I think I do understand a lot of the feelings you both have right now of being overwhelmed. After a while we figured out that counseling would be beneficial, and we took advantage of that. I was diagnosed with depression and it helped to know that that's why I could barely sort through the mail and recycling in the course of a whole day. I found ways to deal with that. But my husband and I also became more of a team and acknowledged each other's fears and problems.