NOAA offers from this link grib files about GFS Forecasts, containing values of forecasted weather attributes. Each file a 3-hour output timestep. Every day is composed of 4 grib files, each one having one of the following time reference; 00:00, 06:00, 12:00, 18:00. The time reference is the time that forecast starts.
This implies that the grib files cover only the following periods of a day; 00:00-03:00, 06:00-09:00, 12:00-03:00, 18:00-21:00.
Are there any other files publicly available, covering the whole day?
No.
New forecast is created every 6 hours, giving new set of files covering hours 0..384. In the archive you are using, it is 3h step for first 240h and 12h step afterwards.
The file pattern is gfs_4_YYYYMMDD_REFT_STEP.grb2, where YYYYMMDD is the date for forecast start time, REFT is hour for forecast start, in CF convention called forecast reference time. STEP is count of hours from the forecast start.
Variables in each file are either instantaneous (like temperature at particular time) or aggregates (precipitation sum over 3h/6h, average radiation over 3h/6h etc.).
So, in order to use data for say the first 12 hours, you should take steps 00, 03, 06, 09 and 12. It is data with 3h interval. Just be careful about the time aggregates in the files, there is a surprise there.