Hi, I'm so sorry for your loss, I've had 3 mc's in 10 months so I feel your pain.
A family member is also an obs & gynae registrar so I do have some advice.
Yes, you can try again straight after, however you are advised to wait for one cycle to help with dating, plus also help your cycle settle down. The problem is, if you catch again before your first period, you have absolutely no way of knowing how many weeks you are because you don't have a last period.
If you therefore have a scan and you are unsure of dates, it can be really distressing to not be sure how far along you are - in that actually you are much earlier than you think but it looks like bad news. Don't put yourself through it.
That said, if you fall pregnant before your first period post mc, it won't harm you or your baby.
My advice would be though, you must wait and get a negative home pregnancy test first and that can be really upsetting. If you don't, you could think you are pregnant again and actually you aren't.
For mc 1, it took me 7 days to get a negative
Mc 2, it took 16 days and for mc 3 it's took 3.5 weeks and to keep seeing a positive line week after week was really distressing.
I know it feels like a lifetime, trust me, I've no dc, I'm 38 and lost 3 babies, and every day that goes past is another day lost.
But waiting until you are back on even ground and starting from fresh is much better. I promise. I understand the desperation to be pregnant again, I truly do.
Big hugs 