One thing to watch out for is the length between the nut and the bridge. 3/4 basses are generally in the region of 105cm. Many 3/4 lower cost basses have shorter playing lengths at anything from around 100 - 105cm because they are aimed at the student market. The shorter string length makes them very playable as you don't have to stretch as much between notes at the lower end of the register ( i.e. equivalent of 1st - 3rd fret on a bass guitar) and the string tension is also less.
The shorter length instruments are a great way to get started but if you want to get a good sound they need a bit more care in terms of fitting correct tension strings otherwise they can sound lifeless and lame ( most strings manufacturers produce data based on how the string will perform at 105/106 cm and logically if you have a shorter instrument of say 100cm, the string doesn't have to stretch as far as the 'design length' and so doesn't 'sing'. Simple way to get around that is to go for a string tension that is 'high' and it will play like a 'medium' on a shorter length instrument. Personally I like Pirastro Obligatos and I have played them on a 103cm bass very successfully with a great pizzicato jazz sound and excellent bowed sound ).
I would definitely consider going second hand and go for a bass comes from a good home in that it is well set up in terms of string height (otherwise it may feel like you are playing 'cheese wire' strings ), and also that the sound post and bridge are correctly positioned. Some new basses can be ok but many have the cheapest set of string on them and need to be set up my an experienced luthier as they come straight out of the factory with a high action so that this can be adjusted to suit the player and their style.
My thoughts would be to get a good 'used' laminate bass which you can pick up anywhere between £500 - £750 and invest in some good strings which may set you back a further £150 ( but you can upgrade the strings at a later date after you have gotten used to the DB !). I say laminate simply because they are more sturdy and robust and ts unlikely that they will develop cracks . When learning its easy to be clumsy carrying the bass around and carved tops (at around this price) can become uneconomical to repair especially if the top needs to come off to stitch a crack in the top ( a student of mine left his Stentor DB in a car overnight, brought it inside the next day into a centrally heated house and got a split below the left f hole!! - you just won't get that problem with a laminate bass ( plywood) which will sound just as good and keep its resale value if you don't bond with the DB as an instrument).
All the best with your quest !
BTW , I have a Boosey and Hawkes instrument that a student has asked me to help find a new home for - ticks all of the above boxes and might suit you really well? If you are interested PM me !