How to sell gift boxes with the ability for the customer to add items

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We are selling gift boxes and would like to allow the customer to add additional items to our standard sets, i am wondering is it possible to do this in basic Shopify or do we need to use an App? The Apps suggested for other questions around custom gift boxes don't look like they will meet our needs. 


Our Eco Friendly box will have as standard 1 babygrow, 1 Bib and a pack of muslin cloths, we would like to have the option then for customer to also add additional items such as an extra babygrow or something not included in the box already like a rattle and / or comforter etc (or they could just go with the standard gift box).

I hope this make sense  

Thanks in advance for your help