There is no need to get a whole new rack, this is a really common problem and they make adapters just for this reason. It connects between the seat post and the stem. They’re really affordable!
Thule Frame Adapter - Bicycle Cross Bar , Black https://www.amazon.com/dp/B004KZS5GW/ref=cm_sw_r_cp_api_glt_fabc_01MA2SPWDA77R6K8RZWT
Looks like it’s for a bike to mount on a tailgate rack.
Like this:
Revgear Steel Cup https://www.amazon.ca/dp/B004KZS5GW/ref=cm_sw_r_cp_api_i_K0NvFbVYW3KZ9
You can also purchase something like this + use a standard bike rack.
If it worries you, one of these adapters would probably solve your problem. There are other brands that are cheaper as well.
https://www.amazon.com/Thule-982XT-Frame-Adapter/dp/B004KZS5GW/