My daughter wiggles all the way to the top of her bassinet with her head against the wood. I have tried swaddling, putting a blanket over her tucked in but nothing seems to prevent this. Is it time to move her to a cot and how can I safely either stop her from doing this or make it more comfortable on her head when she does do it?

What you are describing is very normal for your baby.
Most babies need transferring to a cot around 4 mths of age.
I would also suspect she is not far from rolling.
This is not something you can stop, but it is now important to (continue to) make her environment as safe as possible.
Red Nose recommends the following:
Move her to a cot & cease wrapping…...still place her on her back for sleep.
Ensure the mattress in the cot is firm, well-fitting & flat.
Ensure there is nothing soft, loose or padded in the cot.

It is not uncommon for babies to be rolling by 4 mths.
Most babies will be rolling over by 5 months. This is very normal as babies grow & become stronger.

Many babies then prefer to sleep on their tummy.

Red Nose has the following information now that your baby is starting to roll.

Give baby extra tummy time to play when awake and supervised as this helps baby to develop stronger neck and upper body muscles which in turn enables them to roll back over.
Make sure that baby is on a firm and well-fitting mattress that is flat (not tilted or elevated).
Make sure that baby’s face and head remains uncovered (do not use lambswool, doonas, pillows, cot bumpers or soft toys)

As babies grow and develop they become very active and learn to roll around the cot.
At this time, continue to put them on the back at the start of sleep time, but let them find their own position of comfort.
By this stage it is not necessary to wake during the night to turn baby over to the back position
https://rednose.com.au/article/what-do-we-do-now-that-a-baby-has-started-to-roll-ove
Do not use any devices designed to keep baby in a particular sleep position.
https://rednose.org.au/article/is-it-safe-to-use-products-designed-to-keep-baby-in-a-particular-position