Perhaps avoid the lowest notes too. Please do correct me if I'm wrong but many early 50s and 60s basslines were played higher up the neck as not only were most commercial playback devices not capable of reproducing those lower frequencies, but recording them could cause the vinyl cutter to jump out of the groove.
This could of course be nonsense, but I'm sure I heard it somewhere.
I play in a band that is centred on exactly the sort of material that the OP describes. If you listen to the recorded stuff from that period the bass line very rarely goes below a low A (Eddie Cochran being an interesting counter-example). Ignoring the vinyl cutter issue, if you want the music to sound good on a transistor radio, with some components of the bass part being audible, there's certainly a sweet zone between that A and perhaps the E or F on the G-string.
I'm mostly an upright player these days but surprisingly I don't find that it works well for the do-wop, MOR and early rock and roll stuff. I've been experimenting with various bass guitars for this stuff. Surprisingly I've largely settled on a Yamaha BBP35 which just seems to sound good in every situation. I do like some foam under the strings at the bridge. Stylistically, the most important thing is to get some bounce to the lines: not quite a swing, but with the notes a bit shorter than you'd expect in 4 beats to the bar, mimicking something of a big swing band bass line. That feel is what gives the music from that era a bit of a light-hearted, good-feeling drive to it. (My $0.02 mind you.)