I have switched back and forth over the years. I currently am playing the T300's also but starting with the 6 iron. I have driver, 3 wood, 5 wood, and 23 degree hybrid. So, what you might consider for your choice might best be based on what shots you mostly play between your 3 wood and 5 iron. Do you need longer, higher shots? if so, go 5 wood. Do you want a little more versatility for rough, fairway bunkers, long punch shots under trees? If so, go hybrid. Either one will suit you well as you do need something to fill that current gap. Welcome to this great game and hope you enjoy it for many years to come and welcome to the TT family.

The pros for hybrid is you can swing it like an iron and “rescue” a shot from the rough. Ball flight tends to be high. Downside, can tend to pull or hook. The pros for fairway are the longer shaft and at the same loft, go further. Downside, generally not as easy to hit from the rough, especially as it gets deeper.
That said, it comes down to 2 camps based on how well each group can hit their preferred style club
I used to swear by a 21 degree fairway 7 wood. I now have 21 and 18 degree hybrids with, for me, more consistent results. Hoping I can fill a gap between driver and 18 hybrid with a fairway wood. To each, their own.

If I was you, I would base my decision on gapping between the 3wood and i5. For example, I hit my 3w approximately 220meters and my 5 iron approximately 170m. Therefore I have 5wood that I hit approximately 200 meters (launch is quite high) and utility 3 that i hit approximately 185.

And what comes to the fairway wood vs hybrid, all I can say is book a fitting and test. For example I can't hit hybrids no matter what and feel a lot more comfortable with fairway wood or utility in my hands.

Set up your bag for gap distances and the primary courses you play. Also, traditional club setups with 3 and 5 woods are no longer the norm. It’s all about how you deliver the club, launch, spin, and carry. You may benefit with more loft to achieve optimal performance. Good luck.

All of the above advice is relevant in your circumstances - indeed, we all have to confront similar issues in our set-up. The important aspect is how far you hit each club rather than the number on the bottom of the head.

I was only going to muse that there is an option that hasn't been mentioned and is often overlooked - the U505.
This is an "iron hybrid" that is available in Regular flex shaft options and is a bit of a multi-purpose club, being a good option of the tee, the fairway and light rough. Have a look - it might not suit you but, on the other hand, it might be just what you are looking for.
JT

PS Yes, I've got a U505 4-iron in my bag!!
JT