Let's remind ourselves that the rudder trim, like any trim, is not a control travel limiting system. Therefore, regardless of it's setting, you will still have the full travel of the control available to you. Thus, a miss trimmed aircraft is not a reason to not be able to control it, it just may take some more muscle.
That said, in a recent test flight I did in a Warrior, I did not find the rudder trim to have a tremondous affect on the rudder forces required, as is generally the case for single engined aircraft with rudder trim.
Unlike a Cessna, the Pipers have direct nosewheel steering. Therefore, if, while moving on the ground, you start applying full pedal, you should end up zigzagging all over the place. (avoid, they'll bring out the breathalyzer when they see you). However, once in suitable flight, and safely between 1.4Vs and Va, there's no reason to not get used to the feel of FULL rudder deflection. You can put all the way in, and hold, while you apply opposite aileron, just to get the feel.
Years ago, I had to check myself out in a Bellanca Viking (300HP, four place low wing). I found directional control during takeoff much more difficult than I expected. I attributed it to torque, as the landing was perfectly fine. Then, I was into a runway, where a more soft field takeoff technique seemed appropriate, and it went perfectly. It turned out that I had had the plane trimmed to the wrong end of the pitch trim "takeoff" range, and it was very sensative to that when solo. I had been wheelbarrowing it with takeoff power. When I trimmed way nose up, and kept it light, things went perfectly. After a technique change (which I should have been doing all the way along anyway), no problem!
I was flying a Twin Otter for the first time in many years, a few weeks back. They have a less than desirable nosewheel steering system - a lot of finess required, and over controlling very easy. If doing a short takeoff by yourself, it pretty well requires three hands. I confirmed with my mentor pilot that once perfectly lined up, and with no particular crosswind, the rudder was enough, and streering was not required. He confirmed that. I was not long before I was doing circuits on a rather short and narrow turf runway (just the perfect place for fun in a Twin Otter!). There were some gentle mountain breezes, but nothing which would alarm me. There I was, however, accelerating down the runway with full left pedal, gently going onto the grass on the right side. I suppose it was not alarming my mentor, as he was not helping. Problem was, at this point, you have to undo something fast. Either let off rudder pressure, so you can get you toe up to the brake (bad idea), let the nose back down, so you can reach the nosewheel steering after all (messy, but doable), or start reducing power on the left side. I elected to pull back a bit of power, as I faintly yelled "help!", and I felt a bit of left brake being applied for me. It worked out fine, and my mentor did not seem alarmed, but I was! He agreed that I had had full pedal in, and that should have been lots, but stuff happens! I have a new respect for those pilots who fly Twin Otters into those 50 foot wide runways in the mountains!
Anyway, I've drifted your thread.... Always satisfy yourself that if you might be needing it, you have FULL control applied, it's there to be used when needed. Designers purposefully assure that what you have is what you need, the plane won't swap ends on you!