PLEASE define what a 4/4 is. Almost every Bass I have seen called a 4/4 was actually a 7/8. I have a Huge Bass in repair now and I thought it was a full sized Bass but it's actually STILL a 7/8 to everyone I spoke with. Look here;
http://www.kensmithbasses.com/Double..._that_bass.htm
I think after we define once agian IF ever possible at all, the actual sizes from 1/2, 5/8, 3/4, 7/8, 4/4, and even 5/4 as I have seen some German Basses called we can THEN answer this Question. For now, I don't know if we are all talking about the same thing. I have a 3/4 English Gilkes now that is louder than the 7/8 Italian I used to have. Louder, but not better sounding. The Gilkes is more even but the Italian had a little more color.
I have a large 3/4 Martini (Italian) that is 'AS' loud but deeper and smoother than my 7/8 Shen (high end Chinese). Now this is listening in a concert hall from the back and comparing projection.
Basses do not necessarly sound the same on top of it as you are playing, 5 feet away or 100 feet away.
Your 3/4 modern Bass (hyrbid is a modern thing) is about 41-42" string length. I dont know so you tell us. The Bass gets deeper usually with a longer string length. 7/8 will add an inch in the modern world and full size is at least 44"-46" string length unless it has some other dimentions.
Older Basses have less and less rules of size as we go back counting the centuries. Also, many old Bass have been cut at the shoulders and neck blocks lowered making these older larger Basses a size or so smaller in string length. Some Bodies are longer from the F-notch to the bottom or from the F-notch to the neck block. This will also affect the Basses length but not always it's size.
I am sorry to make this so long but it's difficult to answer a question that has an un-defined variable. I recently bought a FULL Sized Case. It fits fine on my 3/4 Martini and Barely on my 7/8 Shen (now being used by my son). The 3/4 case they make only fits small sloaping shouldered commercial type Basses or Chinese but none with bigger dimentions. Cases used to be bigger but with so many new Basses on the Market the Sizes are NOW re-named to fit what is Current.
Louder with a Bow (or more presence) or Louder with Pizz. This can Vary. If you are going to use an AMP, the smaller Bass will be easier to EQ. My Martini and Gilkes almost kill a speaker when I play the E string as the Fundamental note is so pure without too much upper harmonics. The Bigger the Bass, the more Organ Like tone you may be approaching.
The most common size for Jazz is a modern 20th century German/Czech type Bass in a 3/4. The most common for Orchestra are the older Italian, English and other old Basses in Larger 3/4 or 7/8 depending on who's holding the tape measure.
So.. get the Bass to fit your needs. Size matters only to a degree. Volume, projection, depth of tone and color of tone are not related. Want all the best on on great Bass? Sell your house!!