A five string bass has been my answer for many years now - albeit an electric bass guitar. That won't help you as you don't want to spend lots of money, even for a string bass.
The option of de-tuning is not a bad one, but will jeopardise the rest of the piece as far as fingering is concerned.Were you using an electric bass guitar, then I'd recommend a 'Detuna', which with a flick, takes E down to D.
The option of playing that D note is a possibility, but it must fit in with the rest of the phrase(s). You can't really be playing a line of music downwards towards that D, only to jump nearly an octave when the D is required.
EDIT: another idea I've used on bass guitars - but would work on string basses is to move the strings up, so there is space where the lowest was for a heavier grade string, which gets tuned to B. So, B E A D. Obviously, this works until higher notes are required, but that's not in every piece.
EXTRA EDIT: there's always the 5ths tuning, which is an octave beow ceollo tuning, as in C G D A, so takes the lowest note down far enough, but needs practice to get used to different fingering. That's probably the best option!