At this moment I don't know any contrib module that fit this use-case but I know a module that may interest you in some way.
The module is the Search API Page Block. Here goes a summary of the module description:
This module exposes Search API Pages as blocks and uses the node's
title as keywords. This allows you to use your existing search results
page to direct users to related content.
Currently, this module is only useful when placing search result
blocks on node pages, since it currently only uses $node->title as the
search keywords. Future development can lead to other uses, including
using Taxonomy terms, Context module, or custom fields to set the
To achieve your goal I think you will need to implement a cutom module that stores the result of the user's search and uses the result list as the content of a block implemented by your custom module.
Maybe you can try another approach by using the views module with exposed filters then use the Views filters populate to merge them in a single text field and using a block to show the view result. This approach will let you easily store the keyword in a variable using the hook_form_alter in conjunction with the variable_set and so on.
Here goes a question about the Views Filter Populate that may be useful:
How to combine 2 exposed filters (String and Taxonomy field)