No, the field does not 'top up'.
You cannot sow the field again until it is empty again.
From the page 6 of the rules:
Grain may be sown in empty fields.
In the side panel:
1 Grain becomes 3, 1 Vegetable becomes 2. Players can use the Sow action to sow several empty fields at once. It is irrelevant whether Grain or Vegetables was sown in the field previously. If it has been completely Harvested, it may be re-Sown.
In the extended example in the side panel:
He cannot plant anything in the third field, because it still contains one Grain.
I generally favour sowing and baking over eating animals - I find it gives much more food in the long run.
The general grain/bake bread strategy goes:
Stage 1: Get at least one field sowed. This takes three actions (Plow, Take grain, Sow). If you can, it's nice to get two fields sowed, but there might be other actions (eg. taking wood) that might be more worthwhile.
Stage 2: At harvest you recieved one grain. Plow and sow another field (two actions). Acquire enough food using other means to feed your family.
Stage 3 and beyond: You are now producing at least two grain each harvest. Build either the Stone or Clay oven. Now when you sow, you sow one of your grains, and bake the other(s). That's an easy five/eight food, which is more or less enough to feed your family for that harvest. Also in stage 3 you should get a vegetable, and sow that. The vegetable is for points only, and remember that you only need four for max points, so don't go overboard.
Compare grain to sheep as a food source
Let's say you are getting every single sheep that comes out.
That's only four sheep in stage 1, three sheep in stage 2, and two sheep in the remaining stages. You'll also get one bonus sheep per stage at harvest.
In the later two round stages, that's a maximum of six food you can get solely from cooking sheep.
Whereas if you have say, 5 fields of grain, you could potentially be producing twenty five food from grain each stage. Admittedly this requires you to be using actions. Remember that if you have two ovens you can use them both in the same bake bread action.