Running and increasing appetite

Just a three mile run shouldn't be enough to exponentially increase appetite, based on my experience at least. And carbs tend to help more with running, I increase my carbs to 45-50% when I'm training for a race when I normally eat around 35% when not training. I stay more energized and have better runs.

But having some basic info of where you're at right now will help in trying to determine what you need:
Age:
Weight:
Daily net cal allotment:
Run frequency:
Macros ratios Carb/Pro/Fat:

And when do you normally eat in relation to when you run?

I'd be interested to know the 'right' answer here as well!

Note - I am not saying that there is any "right" or "wrong" answers to questions like this. Every person is different and will ultimately need to discover what works best for them. All I can offer is advice based on my experiences with running, calories, and macros. But there are people out there are are Atkins followers that run and weight lifters that are vegan, so nothing is set in stone in any respect when it comes to stuff like this.

With that said, at 145lbs, if you're average height, I'd guess that you're already in a healthy BMI and I can see from your ticker that you're trying to get down to 125. So the first thing that comes to mind is that that weight may be too low of a goal to get to and maintain if you're over 5'3", it's going to be a struggle to get those last few pounds especially, and likely very frustrating to try to maintain.

It also means that you should be netting more than 1200 calories. The lower your BMI and the less weight you have to lose the smaller your deficit should be. I would recommend upping your net calories by about 100 calories each week for a few weeks until you get up around 1600 or 1700 calories. (if you do it all at once you may see a gain just because your body is trying to readjust and overcompensates)

4 times a week of 3 miles is a good amount for general working out and heart health, but no, that alone shouldn't be making you really hungry. So that's another reason I suspect that raising your net calories will help you out in that respect.

As for the running on an empty stomach, that's a personal preference. I personally hate it and tend to bonk my runs if I've got a totally empty gut. I need something to fuel me or I feel weak and queasy when I'm running, even short distances. For a three mile run in the morning, something simple and not to heavy will be fine. But, like you, I definitely can't run too close to eating most times.

For morning runs, I'm a fan of eggs and toast, maybe with some Greek yogurt beforehand. Then I give myself about an hour for that to settle before I head out and run. Or, if I don't have an hour, I'll go for something like a Luna Bar or Balance Bare bar. I can eat one of those and run pretty much immediately without feeling gross from the food or feeling weak from lack of food. For evening runs, I typically have about two hours after my last full meal before I'll head out.

And for the amount you're running, you don't need to go hugely carb heavy. But yes, it is probably better to not lean quite so heavily to the protein and allow yourself some more carbs. It will feel a lot better during your runs. I kind of alternate ratios a bit from day to day, more carbs on run days, more protein on lifting days and I find that it helps keep me feeling good for both workout types.

But definitely when I trying to do low carbs overall, even before I got into longer distances, I was much more tired and hungry. increasing calories and upping carbs are what stopped that for me.