Is there a way to make fzero reuse values from interval search

This post is half question, half update request...

When calling fzero with only a starting point, not an interval, is there a way to make fzero reuse/recycle the values it has calculated during the initial interval search. I'm guessing not, however the function I'm finding the root of is expensive, and fzero takes ages to execute. See the example output below. At the last iteration of the interval search, fzero has come very close to having the correct answer with a=102.885 and f(a)=79.5417. However, after an interval is identified, fzero then begins the zero search in the interval [99.1788, 120.778]. Surely it would make more sense to use the interval [99.1788, 102.885] because this brackets the solution much more closely. Or is using a smaller interval not possible because the underlying numerical method uses three points.

Search for an interval around 111.831 containing a sign change:
 Func-count    a          f(a)             b          f(b)        Procedure
    1         111.831   1.48227e+06       111.831   1.48227e+06   initial interval
    3         108.668        956222       114.994   2.00547e+06   search
    5         107.358        737493       116.304   2.22137e+06   search
    7         105.505        427326       118.157   2.52586e+06   search
    9         102.885       79.5417       120.778   2.95484e+06   search
   10         99.1788       -626531       120.778   2.95484e+06   search
 
Search for a zero in the interval [99.1788, 120.778]:
 Func-count    x          f(x)             Procedure
   10         99.1788       -626531        initial
   11         102.957       12343.2        interpolation
   12         102.957       12343.2        interpolation
 
Zero found in the interval [99.1788, 120.778]
Solver terminated with area 102.9574