Could also be your tune. An engine runs differently cold than hot, when it's cold you have high idle and choke, that can affect how it starts compared to when it's warm. When it's warm the carb is operating differently and if you are a bit out of time, or if the mixture is not great it can really change your ability to get it started. I agree that since it sounds/acts like battery that it could easily be starter/solenoid/battery cable but remember a car that sits needs regular starts and adjustments, you will get buildup in your carburetor, and fuel lines and crap on your plugs. Check everything since she's been sitting, cleaning and re-gapping your plugs, check the timing, take her for a cruse on the highway at 65-70 blow that crap out of her systems. It will do a world of good.