Don't expect eggs from molting birds until they have completely regrown a new winter coat. There will be no ovulation while they are using all available protein to regrow feathers and the oviduct is getting a much needed rest. Therefor there will be no egg song, no nest box exploration, no need for that reproduction activity. The combs and wattles will pale and shrink. The space between the pointy pelvic bones will decrease because no eggs will need to pass.
One thing you can do, AFTER they have recovered from molt is to add a light on a timer to come on before dawn and gradually increase perceived day length.
The light needs to be bright enough to read a newspaper by at roost height. Start by setting the timer to come on an hour before dawn. Add an hour each week till you are over 12 and up to a total of 14 hours including the ambient light.
Another thing you can do is to cull all of them and start with new pullets every year.
That's what commercial egg farms do. This isn't precisely how everybody does it but it gives you a glimpse into how grocery store eggs are produced and why you may find that the endeavor is no longer worth the effort.
Every month or two, commercial egg companies hatch several hundred thousand new layers and cull all the males as day olds. They rotate flocks continuously during the year. They have large houses for each age of pullets in black out housing. They strictly control day length by keeping them on about 8+ hours of light a day. Feed is about 18% protein till around 14 weeks, then down to about 14-15%. At about 17 weeks, they start a pre-lay diet, perhaps 17% protein and 2.5% calcium. At the same time they start incrementing day length. By about 20 weeks, they are up to 14-16 hours of light. This will kick start all the same breed and same age of pullets to start laying within a week of each other. They have moved them into the layer houses by now - a couple weeks after the previous birds have been culled and housing cleaned and disinfected. They will live in these cages with about 20,000 other pullets with long day length for about 18 months until they are culled just as production begins to tail off. They will be replaced a couple weeks later by the next batch. They have many buildings on each farm that rotate flocks in the same manner.
By artificially controlling light exposure, they simulate spring and long summers for each house regardless of the time of year.
They make money on eggs because they don't waste housing on non-producing birds. They have their own feed mills, feed formulas and buy ingredients by the trainload. This economy of scale makes feeding much cheaper than buying by the bag. About 3 or 4 AM every morning, feed trucks start arriving at the mills to carry feed to the various farms of hen houses. The trucks run all day. The mills run 24/7 except for about 8 hours of maintenance on Sunday evenings. If an automation problem or major malfunction causes the mill to shut down for a day midweek, they're starving chickens, turkeys, etc..
You can do the same thing on a much smaller scale - except you are buying feed in 50# bags, not by the trainload. People with various breeds, whether the same age or not and don't have a lighting program, will never get that assembly line type egg reliability.
Buy or hatch new birds early each year so you have some coming into lay as the older birds are beginning to molt. Cull the older birds then or keep them till after winter solstice and the older birds will start laying slightly larger eggs like gangbusters again till the following autumn when the cycle starts all over.
Increasing day length vis a vis darkness is a signal to all animals to reproduce. Shorter days is a signal to shut it down.
It is difficult to save money or even break even with chickens unless you have a lot of highly productive breeds. That would be breeds like Leghorns, RIRs, New Hampshires, Sussex, Minorcas, Penedesencas, Jaerhons, Hamburgs, Faverolles, Anconas, Catalanas, Aorps, Andalusians.
Orps and Ameraucanas are fairly productive but not as good as the previously mentioned breeds. At any rate, you should get 180-200 eggs per bird per year out of those two breeds if they're fed properly and healthy.
IMO, it is important for us to remember, when they are laying like gangbusters that they still aren't egg laying machines but rather livestock that happen to feed us breakfast and will take periods of time off to rest and rejuvenate.
More pet type breeds mostly lay even more poorly.
